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14.5: Binomial Theorem

14.5: Binomial Theorem


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Learning Objectives

By the end of this section, you will be able to:

  • Use Pascal’s Triangle to expand a binomial
  • Evaluate a binomial coefficient
  • Use the Binomial Theorem to expand a binomial

Before you get started, take this readiness quiz.

  1. Simplify: (frac{7 cdot 6 cdot 5 cdot 4}{4 cdot 3 cdot 2 cdot 1}).
    If you missed this problem, review Example 1.25.
  2. Expand: ((3 x+5)^{2}).
    If you missed this problem, review Example 5.32.
  3. Expand: ((x-y)^{2}).
    If you missed this problem, review Example 5.32.

Use Pascal's Triangle to Expand a Binomial

In our previous work, we have squared binomials either by using FOIL or by using the Binomial Squares Pattern. We can also say that we expanded ((a+b)^{2}).

((a+b)^{2}=a^{2}+2 a b+b^{2})

To expand ((a+b)^{3}), we recognize that this is ((a+b)^{2}(a+b)) and multiply.

((a+b)^{3})
((a+b)^{2}(a+b))
(left(a^{2}+2 a b+b^{2} ight)(a+b))
(a^{3}+2 a^{2} b+a b^{2}+a^{2} b+2 a b^{2}+b^{3})
(a^{3}+3 a^{2} b+3 a b^{2}+b^{3})
((a+b)^{3}=a^{3}+3 a^{2} b+3 a b^{2}+b^{3})

To find a method that is less tedious that will work for higher expansions like ((a+b)^{7}), we again look for patterns in some expansions.

Number of TermsFirst TermLast Term
((a+b)^{1}=a+b)(2)(a^{1})(b^{1})
((a+b)^{2}=a^{2}+2 a b+b^{2})(3)(a^{2})(b^{2})
((a+b)^{3}=a^{3}+3 a^{2} b+3 a b^{2}+b^{3})(4)(a^{3})(b^{3})
((a+b)^{4}=a^{4}+4 a^{3} b+6 a^{2} b^{2}+4 a b^{3}+b^{4})(5)(a^{4})(b^{4})
((a+b)^{5}=a^{5}+5 a^{4} b+10 a^{3} b^{2}+10 a^{2} b^{3}+5 a b^{4}+b^{5})(6)(a^{5})(b^{5})
((a+b)^{n})(n)(a^{n})(b^{n})
Table 12.4.1

Notice the first and last terms show only one variable. Recall that (a^{0}=1), so we could rewrite the first and last terms to include both variables. For example, we could expand ((a+b)^{3}) to show each term with both variables.

Generally, we don’t show the zero exponents, just as we usually write (x) rather than (1x).

Note

Patterns in the Expansion of ((a+b)^{n})

  • The number of terms is (n+1).
  • The first term is (a^{n}) and the last term is (b^{n}).
  • The exponents on (a) decrease by one on each term going left to right.
  • The exponents on (b) increase by one on each term going left to right.
  • The sum of the exponents on any term is (n).

Let’s look at an example to highlight the last three patterns.

From the patterns we identified, we see the variables in the expansion of ((a+b)^{n}), would be

((a+b)^{n}=a^{n}+\_\_\_a^{n-1}b^{1}+\_\_\_a^{n-2}b^{2}+ldots+\_\_\_a^{1}b^{n-1}+b^{n}).

To find the coefficients of the terms, we write our expansions again focusing on the coefficients. We rewrite the coefficients to the right forming an array of coefficients.

The array to the right is called Pascal’s Triangle. Notice each number in the array is the sum of the two closest numbers in the row above. We can find the next row by starting and ending with one and then adding two adjacent numbers.

This triangle gives the coefficients of the terms when we expand binomials.

Definition (PageIndex{1})

Pascal's Triangle

In the next example, we will use this triangle and the patterns we recognized to expand the binomial.

Example (PageIndex{1})

Use Pascal's Triangle to expand ((x+y)^{6}).

Solution:

We know the variables for this expansion will follow the pattern we identified. The nonzero exponents of (x) will start at six and decrease to one. The nonzero exponents of (y) will start at one and increase to six. The sum of the exponents in each term will be six. In our pattern, (a=x) and (b=y).

(egin{array}{l}{(a+b)^{n}=a^{n}+\_\_\_a^{n-1} b^{1}+\_\_\_a^{n-2} b^{2}+ldots+\_\_\_a^{1}b^{n-1}+b^{n}} {(x+y)^{6}=x^{6}+\_\_\_x^{5} y^{1}+\_\_\_x^{4} y^{2}+\_\_\_x^{3} y^{3}+\_\_\_x^{2} y^{4}+\_\_\_x^{1} y^{5}+y^{6}}end{array})

Exercise (PageIndex{1})

Use Pascal’s Triangle to expand ((x+y)^{5}).

Answer

(egin{array}{l}{x^{5}+5 x^{4} y+10 x^{3} y^{2}+10 x^{2} y^{3}} {+5 x y^{4}+y^{5}}end{array})

Exercise (PageIndex{2})

Use Pascal’s Triangle to expand ((p+q)^{7}).

Answer

(egin{array}{c}{p^{7}+7 p^{6} q+21 p^{5} q^{2}+35 p^{4} q^{3}} {+35 p^{3} q^{4}+21 p^{2} q^{5}+7 p q^{6}+q^{7}}end{array})

In the next example we want to expand a binomial with one variable and one constant. We need to identify the (a) and (b) to carefully apply the pattern.

Example (PageIndex{2})

Use Pascal’s Triangle to expand ((x+3)^{5}).

Solution:

We identify the (a) and (b) of the pattern.

In our pattern, (a=x) and (b=3).

We know the variables for this expansion will follow the pattern we identified. The sum of the exponents in each term will be five.

((a+b)^{n}=a^{n}+\_\_\_a^{n-1}b^{1}+\_\_\_a^{n-2}b^{2}+ldots+\_\_\_a^{1}b^{n-1}+b^{n} )

((x+3)^{5}=x^{5}+\_\_\_x^{4}cdot3^{1}+\_\_\_x^{3}cdot3^{2}+\_\_\_x^{2}cdot3^{3}+\_\_\_x^{1}cdot3^{4}+3^{5})

Exercise (PageIndex{3})

Use Pascal's Triangle to expand ((x+2)^{4}).

Answer

(x^{4}+8 x^{3}+24 x^{2}+32 x+16)

Exercise (PageIndex{4})

Use Pascal's Triangle to expand ((x+1)^{6}).

Answer

(egin{array}{l}{x^{6}+6 x^{5}+15 x^{4}+20 x^{3}+15 x^{2}} {+6 x+1}end{array})

In the next example, the binomial is a difference and the first term has a constant times the variable. Once we identify the (a) and (b) of the pattern, we must once again carefully apply the pattern.

Example (PageIndex{3})

Use Pascal's Triangle to expand ((3x-2)^{4}).

Solution:

We identify the (a) and (b) of the pattern.

In our pattern, (a=3x) and (b=-2).

((a+b)^{n}=a^{n}+\_\_\_a^{n-1}b^{1}+\_\_\_a^{n-2}b^{2}+ldots+\_\_\_a^{1}b^{n-1}+b^{n} )

((3 x-2)^{4}=81 x^{4}+4left(27 x^{3} ight)(-2)+6left(9 x^{2} ight)(4)+4(3 x)(-8)+1 cdot 16)

Exercise (PageIndex{5})

Use Pascal's Triangle to expand ((2x-3)^{4}).

Answer

(16 x^{4}-96 x^{3}+216 x^{2}-216 x+81)

Exercise (PageIndex{6})

Use Pascal's Triangle to expand ((2x-1)^{6}).

Answer

(egin{array}{l}{64 x^{6}-192 x^{5}+240 x^{4}-160 x^{3}} {+60 x^{2}-12 x+1}end{array})

Evaluate a Binomial Coefficient

While Pascal’s Triangle is one method to expand a binomial, we will also look at another method. Before we get to that, we need to introduce some more factorial notation. This notation is not only used to expand binomials, but also in the study and use of probability.

To find the coefficients of the terms of expanded binomials, we will need to be able to evaluate the notation (left( egin{array}{l}{n} {r}end{array} ight)) which is called a binomial coefficient. We read (left( egin{array}{l}{n} {r}end{array} ight)) as “(n) choose (r)” or “(n) taken (r) at a time”.

Definition (PageIndex{1})

A binomial coefficient (left( egin{array}{l}{n} {r}end{array} ight)), where (r) and (b) are integers with (0 leq r leq n), is defined as

(left( egin{array}{l}{n} {r}end{array} ight)=frac{n !}{r !(n-r) !})

We read (left( egin{array}{l}{n} {r}end{array} ight)) as "(n) choose (r)" or "(n) taken (r) at a time."

Example (PageIndex{4})

Evaluate:

  1. (left( egin{array}{l}{5} {1}end{array} ight))
  2. (left( egin{array}{l}{7} {7}end{array} ight))
  3. (left( egin{array}{l}{4} {0}end{array} ight))
  4. (left( egin{array}{l}{8} {5}end{array} ight))

Solution:

a. We will use the definition of a binomial coefficient,

(left( egin{array}{l}{n} {r}end{array} ight)=frac{n !}{r !(n-r) !})

(left( egin{array}{l}{5} {1}end{array} ight))

Use the definition, (left( stackrel{5}{1} ight)=frac{n !}{r !(n-r) !}), where (n=5, r=1).

(frac{5 !}{1 !(5-1) !})

Simplify.

(frac{5 !}{1 !(4) !})

Rewrite (5!) as (5cdot 4!)

(frac{5 cdot 4 !}{1 ! cdot 4 !})

Simplify, by removing common factors.

(frac{5cdot cancel{4 !}}{1 ! cdot cancel{4 !}})

Simplify.

(5)

(left( egin{array}{l}{5} {1}end{array} ight)=5)

b. (left( egin{array}{l}{7} {7}end{array} ight))

Use the definition, (left( stackrel{5}{1} ight)=frac{n !}{r !(n-r) !}), where (n=7, r=7).

(frac{7 !}{7 !(7-7) !})

Simplify.

(frac{7 !}{7 !(0) !})

Simplify. Remember (0!=1).

(1)

(left( egin{array}{l}{7} {7}end{array} ight)=1)

c. (left( egin{array}{l}{4} {0}end{array} ight))

Use the definition, (left( stackrel{5}{1} ight)=frac{n !}{r !(n-r) !}), where (n=4, r=0).

(frac{4 !}{0 !(4-0) !})

Simplify.

(frac{4 !}{0 !(4) !})

Simplify.

(1)

(left( egin{array}{l}{4} {0}end{array} ight)=1)

d. (left( egin{array}{l}{8} {5}end{array} ight))

Use the definition, (left( stackrel{5}{1} ight)=frac{n !}{r !(n-r) !}), where (n=8, r=5).

(frac{8 !}{5 !(8-5) !})

Simplify.

(frac{8 !}{5 !(3) !})

Rewrite (8!) as (8cdot 7cdot 6cdot 5!) and remove common factors.

(frac{8cdot7cdotcancel{6}cdotcancel{5!}}{cancel{5!}cdotcancel{3}cdotcancel{2}cdot1})

Simplify.

(56)

(left( egin{array}{l}{8} {5}end{array} ight)=56)

Exercise (PageIndex{7})

Evaluate each binomial coefficient:

  1. (left( egin{array}{l}{6} {1}end{array} ight))
  2. (left( egin{array}{l}{8} {8}end{array} ight))
  3. (left( egin{array}{l}{5} {0}end{array} ight))
  4. (left( egin{array}{l}{7} {3}end{array} ight))
Answer
  1. (6)
  2. (1)
  3. (1)
  4. (35)

Exercise (PageIndex{8})

Evaluate each binomial coefficient:

  1. (left( egin{array}{l}{2} {1}end{array} ight))
  2. (left( egin{array}{l}{11} {11}end{array} ight))
  3. (left( egin{array}{l}{9} {0}end{array} ight))
  4. (left( egin{array}{l}{6} {5}end{array} ight))
Answer
  1. (2)
  2. (1)
  3. (1)
  4. (6)

In the previous example, ((a)), ((b)), ((c)) demonstrate some special properties of binomial coefficients.

Definition (PageIndex{2})

Properties of Binomial Coefficients

(left( egin{array}{l}{n} {1}end{array} ight)=n quad left( egin{array}{l}{n} {n}end{array} ight)=1 quad left( egin{array}{l}{n} {0}end{array} ight)=1)

Use the Binomial Theorem to Expand a Binomial

We are now ready to use the alternate method of expanding binomials. The Binomial Theorem uses the same pattern for the variables, but uses the binomial coefficient for the coefficient of each term.

Definition (PageIndex{3})

Binomial Theorem

For any real numbers (a) and (b), and positive integer (n),

((a+b)^{n}=left( egin{array}{c}{n} {0}end{array} ight) a^{n}+left( egin{array}{c}{n} {1}end{array} ight) a^{n-1} b^{1}+left( egin{array}{c}{n} {2}end{array} ight) a^{n-2} b^{2}+ldots+left( egin{array}{c}{n} {r}end{array} ight) a^{n-r} b^{r}+ldots+left( egin{array}{c}{n} {n}end{array} ight) b^{n})

Example (PageIndex{5})

Use the Binomial Theorem to expand ((p+q)^{4}).

Solution:

We identify the (a) and (b) of the pattern.

In our pattern, (a=p) and (b=q).

We use the Binomial Theorem.

((a+b)^{n}=left( egin{array}{c}{n} {0}end{array} ight) a^{n}+left( egin{array}{c}{n} {1}end{array} ight) a^{n-1} b^{1}+left( egin{array}{c}{n} {2}end{array} ight) a^{n-2} b^{2}+ldots+left( egin{array}{c}{n} {r}end{array} ight) a^{n-r} b^{r}+ldots+left( egin{array}{c}{n} {n}end{array} ight) b^{n})

Substitute in the values (a=p, b=q) and (n=4).

((p+q)^{4}=left( egin{array}{c}{4} {0}end{array} ight) p^{4}+left( egin{array}{c}{4} {1}end{array} ight) p^{4-1} q^{1}+left( egin{array}{c}{4} {2}end{array} ight) p^{4-2} q^{2}+left( egin{array}{c}{4} {3}end{array} ight) p^{4-3} q^{3}+left( egin{array}{c}{4} {4}end{array} ight) q^{4})

Simplify the exponents.

((p+q)^{4}=left( egin{array}{l}{4} {0}end{array} ight) p^{4}+left( egin{array}{c}{4} {1}end{array} ight) p^{3} q+left( egin{array}{c}{4} {2}end{array} ight) p^{2} q^{2}+left( egin{array}{c}{4} {3}end{array} ight) p q^{3}+left( egin{array}{c}{4} {4}end{array} ight) q^{4})

Evaluate the coefficients, remember, (left( egin{array}{l}{n} {1}end{array} ight)=n, left( egin{array}{l}{n} {n}end{array} ight)=1, left( egin{array}{l}{n} {0}end{array} ight)=1)

((p+q)^{4}=1 p^{4}+4 p^{3} q^{1}+frac{4 !}{2 !(2) !} p^{2} q^{2}+frac{4 !}{3 !(4-3) !} p^{1} q^{3}+1 q^{4})
((p+q)^{4}=p^{4}+4 p^{3} q+6 p^{2} q^{2}+4 p q^{3}+q^{4})

Exercise (PageIndex{9})

Use the Binomial Theorem to expand ((x+y)^{5}).

Answer

(egin{array}{l}{x^{5}+5 x^{4} y+10 x^{3} y^{2}+10 x^{2} y^{3}} {+5 x y^{4}+y^{5}}end{array})

Exercise (PageIndex{10})

Use the Binomial Theorem to expand ((m+n)^{6}).

Answer

(egin{array}{l}{m^{6}+6 m^{5} n+15 m^{4} n^{2}+20 m^{3} n^{3}} {+15 m^{2} n^{4}+6 m n^{5}+n^{6}}end{array})

Notice that when we expanded ((p+q)^{4}) in the last example, using the Binomial Theorem, we got the same coefficients we would get from using Pascal's Triangle.

The next example, the binomial is a difference. When the binomial is a difference, we must be careful in identifying the values we will use in the pattern.

Example (PageIndex{6})

Use the Binomial Theorem to expand ((x-2)^{5}).

Solution:

We identify the (a) and (b) of the pattern.

In our pattern, (a=x) and (b=-2).

We use the Binomial Theorem.

((a+b)^{n}=left( egin{array}{c}{n} {0}end{array} ight) a^{n}+left( egin{array}{c}{n} {1}end{array} ight) a^{n-1} b^{1}+left( egin{array}{c}{n} {2}end{array} ight) a^{n-2} b^{2}+ldots+left( egin{array}{c}{n} {r}end{array} ight) a^{n-r} b^{r}+ldots+left( egin{array}{c}{n} {n}end{array} ight) b^{n})

Substitute in the values (a=x, b=-2), and (n=5).

((x-2)^{5}=left( egin{array}{l}{5} {0}end{array} ight) x^{5}+left( egin{array}{c}{5} {1}end{array} ight) x^{5-1}(-2)^{1}+left( egin{array}{c}{5} {2}end{array} ight) x^{5-2}(-2)^{2}+left( egin{array}{c}{5} {3}end{array} ight) x^{5-3}(-2)^{3}+left( egin{array}{c}{5} {4}end{array} ight) x^{5-4}(-2)^{4}+left( egin{array}{c}{5} {5}end{array} ight)(-2)^{5})

Simplify the coefficients. Remember, (left( egin{array}{l}{n} {1}end{array} ight)=n, left( egin{array}{l}{n} {n}end{array} ight)=1, left( egin{array}{l}{n} {0}end{array} ight)=1).

((x-2)^{5}=left( egin{array}{l}{5} {0}end{array} ight) x^{5}+left( egin{array}{c}{5} {1}end{array} ight) x^{4}(-2)+left( egin{array}{c}{5} {2}end{array} ight) x^{3}(-2)^{2}+left( egin{array}{c}{5} {3}end{array} ight) x^{2}(-2)^{3}+left( egin{array}{c}{5} {4}end{array} ight) x(-2)^{4}+left( egin{array}{c}{5} {5}end{array} ight)(-2)^{5})

((x-2)^{5}=1 x^{5}+5(-2) x^{4}+frac{5 !}{2 ! cdot 3 !}(-2)^{2} x^{3}+frac{5 !}{3 ! 2 !}(-2)^{3} x^{2}+frac{5 !}{4 !1 !}(-2)^{4} x+1(-2)^{5})

((x-2)^{5}=x^{5}+5(-2) x^{4}+10 cdot 4 cdot x^{3}+10(-8) x^{2}+5 cdot 16 cdot x+1(-32))

Exercise (PageIndex{11})

Use the Binomial Theorem to expand ((x-3)^{5}).

Answer

(egin{array}{l}{x^{5}-15 x^{4}+90 x^{3}-270 x^{2}} {+405 x-243}end{array})

Exercise (PageIndex{12})

Use the Binomial Theorem to expand ((y-1)^{6}).

Answer

(egin{array}{l}{y^{6}-6 y^{5}+15 y^{4}-20 y^{3}+15 y^{2}} {-6 y+1}end{array})

Things can get messy when both terms have a coefficient and a variable.

Example (PageIndex{7})

Use the Binomial Theorem to expand ((2x-3y)^{4}).

Solution:

We identify the (a) and (b) of the pattern.

In our pattern, (a=2x) and (b=-3y).

We use the Binomial Theorem.

((a+b)^{n}=left( egin{array}{c}{n} {0}end{array} ight) a^{n}+left( egin{array}{c}{n} {1}end{array} ight) a^{n-1} b^{1}+left( egin{array}{c}{n} {2}end{array} ight) a^{n-2} b^{2}+ldots+left( egin{array}{c}{n} {r}end{array} ight) a^{n-r} b^{r}+ldots+left( egin{array}{c}{n} {n}end{array} ight) b^{n})

Substitute in the values (a=2x, b=-3y) and (n=4).

Simplify the exponents.

Evaluate the coefficients. Remember, (left( egin{array}{l}{n} {1}end{array} ight)=n, left( egin{array}{l}{n} {n}end{array} ight)=1, left( egin{array}{l}{n} {0}end{array} ight)=1)

Exercise (PageIndex{13})

Use the Binomial Theorem to expand ((3x-2y)^{5}).

Answer

(egin{array}{l}{243 x^{5}-810 x^{4} y+1080 x^{3} y^{2}} {-720 x^{2} y^{3}+240 x y^{4}-32 y^{5}}end{array})

Exercise (PageIndex{14})

Use the Binomial Theorem to expand ((4x-3y)^{4}).

Answer

(egin{array}{l}{256 x^{4}-768 x^{3} y+864 x^{2} y^{2}} {-432 x y^{3}+81 y^{4}}end{array})

The real beauty of the Binomial Theorem is that it gives a formula for any particular term of the expansion without having to compute the whole sum. Let’s look for a pattern in the Binomial Theorem.

Notice, that in each case the exponent on the (b) is one less than the number of the term. The ((r+1)^{st}) term is the term where the exponent of (b) is (r). So we can use the format of the ((r+1)^{st}) term to find the value of a specific term.

Note

Find a Specific Term in a Binomial Expansion

The ((r+1)^{s t}) term in the expansion of ((a+b)^{n}) is

(left( egin{array}{l}{n} {r}end{array} ight) a^{n-r} b^{r})

Example (PageIndex{8})

Find the fourth term of ((x+y)^{7}).

Solution:

In our pattern, (n=7, a=x) and (b=y).

We are looking for the fourth term.

Since (r+1=4), then (r=3).

Write the formula
Substitute in the values, (n=7, r=3, a=x), and (b=y).
Simplify.
Simplify.
Table 12.4.1

Exercise (PageIndex{15})

Find the third term of ((x+y)^{6}).

Answer

(15x^{4}y^{2})

Exercise (PageIndex{16})

Find the fifth term of ((a+b)^{8}).

Answer

(8ab^{7})

Exercise (PageIndex{17})

Find the coefficient of the (x^{5}) term of ((x+4)^{8}).

Answer

(7,168)

Exercise (PageIndex{18})

Find the coefficient of the (x^{4}) term of ((x+2)^{7}).

Answer

(280)

Access these online resources for additional instruction and practice with sequences.

  • Binomial Expansion Using Pascal’s Triangle
  • Binomial Coefficients

Key Concepts

  • Patterns in the expansion of ((a+b)^{n}(
    • The number of terms is (n+1).
    • The first term is (a^{n}) and the last term is (b^{n}).
    • The exponents on (a) decrease by one on each term going left to right.
    • The exponents on (b) increase by one on each term going left to right.
    • The sum of the exponents on any term is (n).
  • Pascal’s Triangle
  • Binomial Coefficient (left( egin{array}{l}{mathbf{n}} {mathbf{r}}end{array} ight)) : A binomial coefficient (left( egin{array}{l}{mathbf{n}} {mathbf{r}}end{array} ight)), where (r) and (n) are integers with (0≤r≤n), is defined as

(left( egin{array}{l}{n} {r}end{array} ight)=frac{n !}{r !(n-r) !})

We read (left( egin{array}{l}{n} {r}end{array} ight)) as “(n) choose (r)” or “(n) taken (r) at a time”.

  • Properties of Binomial Coefficients

(left( egin{array}{l}{n} {1}end{array} ight)=n quad left( egin{array}{l}{n} {n}end{array} ight)=1 quad left( egin{array}{l}{n} {0}end{array} ight)=1)

  • Binomial Theorem:

For any real numbers (a), (b), and positive integer (n),

((a+b)^{n}=left( egin{array}{c}{n} {0}end{array} ight) a^{n}+left( egin{array}{c}{n} {1}end{array} ight) a^{n-1} b^{1}+left( egin{array}{c}{n} {2}end{array} ight) a^{n-2} b^{2}+ldots+left( egin{array}{c}{n} {r}end{array} ight) a^{n-r} b^{r}+ldots+left( egin{array}{c}{n} {n}end{array} ight) b^{n})


14.5: Binomial Theorem

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1.3 Deductive Reasoning 3.3 Ways of Stating a Conditional
3.3 Define Converse, Inverse, & Contrapositive
3.1 Statements 3.3 Equivalence - Con., Inv. & Contra. (Method 1) (Method 2)
3.1 Connectives 3.3 More Converse, Inverse & Contrapositive Examples
3.1 More Samples with Connectives 3.3 Biconditional and Equivalence (Method 1) (Method 2)
3.1 Quantifiers 3.3 Implication Property (Method 1) (Method 2)
3.1 Negating Quantified Statements 3.3 Negation of Sufficient and Necessary
3.1 Consistency Between Quantified Statements
3.4 Law of Detachment & Fallacy of Converse (Md1) (Md2)
3.2 Truth Tables - Negation, Conjunction, Disjunction 3.4 Law of Contraposition & Fallacy of Inverse (Md1) (Md2)
3.2 Constructing Truth Tables (Method 1) (Method 2) 3.4 Disjunctive Syllogism (Method 1) (Method 2)
3.2 Number of Cases for a Truth Table 3.4 Hypothetical Syllogism (Method 1) (Method 2)
3.2 Tautology & Contradiction (Method 1) (Method 2) 3.4 Truth Table for Complex Arguments (Method 1) (Method 2)
3.2 Equivalence - Distributive (Method 1) (Method 2) 3.4 T-Proof for an Argument
3.2 DeMorgan's Laws (Method 1) (Method2) 3.4 Another T-Proof for an Argument
3.5 Syllogisms with Universal & Existential Quantifiers
Summary - "Real" Arguments 3.5 More Syllogisms with Universal & Existential Quantifiers

Chapter Thirteen Counting

13.1 Introduction to Counting Methods 13.3 Permutations
13.3 Permutation Formulas
13.2 Fundamental Counting Principle 13.3 Permutations with Repetitions
13.2 Fundamental Counting Principle (More Examples) 13.3 Combinations
13.3 Permutations and Combinations
13.4 Poker
13.4 Powerball

Chapter Fourteen Probability

Chapter Fifteen Descriptive Statistics

15.1 Surveys and Data Collection 15.3 Measures of Dispersion
15.1 Basic Definitions for Statistics 15.3 More Measures of Dispersion
15.1 Frequency and Relative Frequency Tables (7:20) 15.3 Coefficient of Variation
15.1 Bar Graphs and Histograms
15.1 Interpreting Histograms 15.4 Introduction to the Normal Curve (7:11)
15.1 Stem-and-Leaf Displays 15.4 Introduction to z-score
15.4 More z-score Examples
15.2 What is "average"? 15.4 More z-score Applications
15.2 Advantages & Disadvantages of Types of Averages
15.2 Compute Mean, Median, Mode, and Midrange 15.5 Line of Best Fit - Motivation - Mile Run
15.2 Inferences Knowing Average and More Info (7:18) 15.5 Compute Line of Best Fit
15.2 Five-Number-Summary and Box-and Whisker 15.5 Compute Linear Correlation Coefficient
15.5 Properties of the Linear Correlation Coefficient
15.5 Summary Example - First Class Stamps

Microsoft Excel 2003 for Statistics

Microsoft Excel 2007 for Statistics

Copyright 200 6– 200 7 Timothy Peil

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5 Binomial Test

The binomial test is used to look for evidence that the proportion of a Binomial distributed random variable may differ from a hypothesized (or previously observed) value.

5.1.1 Test Statistic

The test statistic for a binomial test is the observed frequency of experimental subjects that exhibit the trait of interest.

5.1.2 Definitions

Let (X) be a random variable following a binomial distribution with parameters (n) and (pi) . Let (x) be the observed frequency of experimental subjects exhibiting the trait of interest.

5.1.3 Hypotheses

The hypotheses for the Binomial test may take the following forms:

[egin H_0: pi = pi_0 H_a: pi eq pi_0 end]

[egin H_0: pi < pi_0 H_a: pi geq pi_0 end]

[egin H_0: pi > pi_0 H_a: pi leq pi_0 end]

5.1.4 Decision Rule

The decision to reject the null hypothesis is made when the observed value of (x) lies in the critical region that suggests the probability of that observation is low. We define the critical region as the upper bound we are willing to accept for (alpha) , the Type I Error.

In a two-sided test, the upper bound is shared equally in both tails. Due to the discrete nature of the distribution, the total probability in the tails may not equal (alpha) . The figures below depict examples of rejection regions for selected values of the Binomial distribution parameters. The decision rule is:

Reject (H_0) if (x < Binomial(alpha/2, n, pi_0)) or (x > Binomial(1 - alpha/2, n, pi_0))

Figure 5.1: The examples displayed use (n = 20) . For the top, middle, and bottom examples, (pi) is set at 0.3, 0.5, and 0.75, respectively. Notice that in some cases, the rejection regions for (alpha = 0.10) and (alpha = 0.05) are identical.

In the one-sided test, (alpha) is placed in only one tail. The figures below depict examples of rejection regions for selected values of the Binomial distribution parameters. In each case, (alpha) is the area in the tail of the figure. It follows, then, that the decision rule for a lower tailed test is:

Reject (H_0) when (x leq Binomial(alpha, n, pi_0))

For an upper tailed test, the decision rule is:

Reject (H_0) when (x geq Binomial(1 - alpha, n, pi_0))

Figure 5.2: The examples displayed use (n = 20) . For the top, middle, and bottom examples, (pi) is set at 0.3, 0.5, and 0.75, respectively.

5.1.5 Power

The derivations below make use of the following symbols:

  • (x) : The observed frequency of experimental units exhibiting the trait of interest.
  • (n) : The total number of experimental units.
  • (pi_0) : The proportion of the population that exhibits the trait of interest under the null hypothesis.
  • (pi_a) : The proportion of the population that exhibits the trait of interest under the alternative hypothesis.
  • (alpha) : The significance level.
  • (gamma(pi)) : The power of the test for the parameter (pi) .
  • (Binomial(alpha, n, pi)) : A quantile of the Binomial distribution with a probability (alpha) , and parameters (n) and (pi) .
  • (C) : The critical region.

Two Sided Test

[egin gamma(pi_a) &= P_(x in C) &= P_(Binomial(alpha/2, n, pi_0) leq Binomial(alpha / 2, n, pi_a)) + & P_(Binomial(1 - alpha / 2, n, pi_0) geq Binomial(1 - alpha / 2, n, pi_a)) end]

Left Sided Test

[egin gamma(pi_a) &= P_(x in C) &= P_(Binomial(alpha, n, pi_0) leq Binomial(alpha, n, pi_a)) end]

Right Sided Test

[egin gamma(pi_a) &= P_(x in C) &= P_(Binomial(1 - alpha, n, pi_0) geq Binomial(1 - alpha, n, pi_a)) end]

Since the Binomial distribution is discrete, the power curve has the interesting characteristic of not being monotonic. It is sometimes described as having a “sawtooth” appearance. This behavior means that a larger sample size is not always preferred. For example, in the following figure, a sample size of 10 has better power than a sample size of 12.

Figure 5.3: Power for a Binomial test with (pi_0 = .15) and (pi_a = 0.25)


About the Author

CARLOS A. BRAUMANN is Professor in the Department of Mathematics and member of the Research Centre in Mathematics and Applications, Universidade de Évora, Portugal. He is an elected member of the International Statistical Institute (since 1992), a former President of the European Society for Mathematical and Theoretical Biology 񢉙ᆠ) and of the Portuguese Statistical Society 񢉖ᆝ and 2009ᆠ), and a former member of the European Regional Committee of the Bernoulli Society 񢉘ᆠ). He has dealt with stochastic differential equation (SDE) models and applications (mainly biological).


33.4 Preliminary Results in the Field of Real Numbers

33.4.1 Theorem: Uniqueness of Identities

Identity elements are unique.

Proof: Suppose (u star a=astar u=a) and (estar a=astar e=a, forall ainRe) . Then (u=ustar e=e) .

33.4.2 Theorem 2: Uniqueness of Inverses

If (star) is an associative operation, inverse elements for (star) are unique.

Suppose (astar a_1=a_1star a=e) and (astar a_2=a_2star a=e) , where (e) is the identity element for the operation (star) .

Then (a_1=a_1star e=a_1star(astar a_2)=(a_1star a)star a_2)=estar a_2=a_2) .

33.4.3 Theorem: Left Cancellation Law

If (a) has an inverse (a^prime) with respect to the associative operation (star) , and (astar b=astar c) , then (b=c) .

Suppose (astar b=astar c) . Then

[egin a^primestar(astar b) &= a^primestar(astar c) Rightarrow(a^primestar a)star b &= (a^primestar a)star c Rightarrow estar b &= estar c Rightarrow b &= c end]

33.4.4 Corollary: Right Cancellation

In the field ((Re,+,cdot), a+b=a+cRightarrow b=c) , and if (a eq 0) and (ab=ac) , then (b=c) .

Proof: The Corollary is proved using commutativity and Left Cancellation.

33.4.5 Lemma

By axiom 4 (-0+0=0) . Because 0 is the additive identity, (-0+0=-0) . Therefore (-0=-0+0=0) .

33.4.6 Theorem

(forall ainRe, a>0) if and only if (ainRe^+) .

Proof: Suppose (0<a) . Then (a-0inRe^+) , but (a-0=a) , so (ainRe^+) . Conversely, suppose (ainRe^+) . Then (a-0=ainRe^+) , so (0<a) .

33.4.7 Theorem

[egin 0 + xcdot 0 &= xcdot 0 &= xcdot(0+0) &= xcdot 0 + xcdot 0 Rightarrow 0 &= xcdot 0 end]


4.3 Expected Values

  1. (q = (1 - pi))
  2. Let (y = x - 1) and (n = m + 1)
    (Rightarrow) (x = y + 1) and (m = n - 1)
  3. (sumlimits_^ypi^yq^) is of the form of the expected value of (Y) , and (E(Y)=mpi=(n-1)pi) . (sumlimits_^pi^yq^) is the sum of all probabilities over the domain of (Y) which is 1.

[egin mu &= E(X) &= npi sigma^2 &= E(X^2) - E(X)^2 &= n^2pi^2 - npi^2 + npi - n^2pi^2 &= -npi^2 + npi &= npi(-pi-1) &= npi(1-pi) end]


About The Basic Practice Of Statistics 8th Edition Pdf Free

David Moore pioneered the “data analysis” (conceptual) approach for The Basic Practice Of Statistics 8th Edition Moore Pdf. Moore’s The Basic Practice Of Statistics Pdf became the market-leading bestseller by focusing on how statistics are gathered, analyzed, and applied to real problems and situations—and by confronting student anxieties about the course’s relevance and difficulties head on.

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  • Chapter 1: Picturing Distributions with Graphs
    • 1.1: Individuals and variables (1)
    • 1.2: Categorical variables: pie charts and bar graphs (3)
    • 1.3: Quantitative variables: histograms (1)
    • 1.4: Interpreting histograms
    • 1.5: Quantitative variables: stemplots (2)
    • 1.6: Time plots (1)
    • 1: Check Your Skills (10)
    • 1: Exercises (19)
    • 1: Exploring the Web
    • 1: Extra Problems (4)
    • 2.1: Measuring center: the mean (2)
    • 2.2: Measuring center: the median
    • 2.3: Comparing the mean and the median (3)
    • 2.4: Measuring variability: the quartiles
    • 2.5: The five-number summary and boxplots (2)
    • 2.6: Spotting suspected outliers and modified boxplots (2)
    • 2.7: Measuring variability: the standard deviation
    • 2.8: Choosing measures of center and variability (3)
    • 2.9: Using technology
    • 2.10: Organizing a statistical problem (2)
    • 2: Check Your Skills (10)
    • 2: Exercises (18)
    • 2: Exploring the Web
    • 2: Extra Problems (12)
    • 3.1: Density curves (2)
    • 3.2: Describing density curves (2)
    • 3.3: Normal distributions
    • 3.4: The 68&ndash95&ndash99.7 rule (2)
    • 3.5: The standard Normal distribution (2)
    • 3.6: Finding Normal proportions
    • 3.7: Using the standard Normal table (3)
    • 3.8: Finding a value given a proportion (2)
    • 3: Check Your Skills (10)
    • 3: Exercises (27)
    • 3: Exploring the Web
    • 3: Extra Problems (19)
    • 4.1: Explanatory and response variables (3)
    • 4.2: Displaying relationships: scatterplots (1)
    • 4.3: Interpreting scatterplots (1)
    • 4.4: Adding categorical variables to scatterplots
    • 4.5: Measuring linear association: correlation (1)
    • 4.6: Facts about correlation (2)
    • 4: Check Your Skills (10)
    • 4: Exercises (18)
    • 4: Exploring the Web
    • 4: Extra Problems (4)
    • 5.1: Regression lines (3)
    • 5.2: The least-squares regression line
    • 5.3: Using technology (2)
    • 5.4: Facts about least-squares regression (1)
    • 5.5: Residuals (3)
    • 5.6: Influential observations (3)
    • 5.7: Cautions about correlation and regression (3)
    • 5.8: Association does not imply causation (3)
    • 5: Check Your Skills (10)
    • 5: Exercises (26)
    • 5: Exploring the Web
    • 5: Extra Problems
    • 6.1: Marginal distributions (2)
    • 6.2: Conditional distributions (3)
    • 6.3: Simpson's paradox (2)
    • 6: Check Your Skills (7)
    • 6: Exercises (14)
    • 6: Exploring the Web
    • 6: Extra Problems
    • 7: Test Yourself (33)
    • 7: Supplementary Exercises (15)
    • 7: Extra Problems (1)
    • 8.1: Population versus sample (3)
    • 8.2: How to sample badly (2)
    • 8.3: Simple random samples (3)
    • 8.4: Inference about the population (1)
    • 8.5: Other sampling designs (2)
    • 8.6: Cautions about sample surveys (1)
    • 8.7: The impact of technology (1)
    • 8: Check Your Skills (7)
    • 8: Exercises (16)
    • 8: Exploring the Web
    • 8: Extra Problems (5)
    • 9.1: Observation versus experiment (3)
    • 9.2: Subjects, factors, and treatments (3)
    • 9.3: How to experiment badly (1)
    • 9.4: Randomized comparative experiments (2)
    • 9.5: The logic of randomized comparative experiments (2)
    • 9.6: Cautions about experimentation (2)
    • 9.7: Matched pairs and other block designs (3)
    • 9: Check Your Skills (9)
    • 9: Exercises (19)
    • 9: Exploring the Web
    • 9: Extra Problems (1)
    • 10.1: Institutional review boards (1)
    • 10.2: Informed consent (1)
    • 10.3: Confidentiality (1)
    • 10.4: Clinical trials
    • 10.5: Behavioral and social science experiments (1)
    • 10: Check Your Skills
    • 10: Exercises (6)
    • 10: Exploring the Web
    • 10: Extra Problems
    • 11: Test Yourself (14)
    • 11: Supplementary Exercises (6)
    • 11: Extra Problems (1)
    • 12.1: The idea of probability
    • 12.2: The search for randomness (3)
    • 12.3: Probability models (3)
    • 12.4: Probability rules (4)
    • 12.5: Finite and discrete probability models (3)
    • 12.6: Continuous probability models (3)
    • 12.7: Random variables (2)
    • 12.8: Personal probability (2)
    • 12: Check Your Skills (10)
    • 12: Exercises (25)
    • 12: Exploring the Web
    • 12: Extra Problems (28)
    • 13.1: Independence and the multiplication rule (3)
    • 13.2: The general addition rule (2)
    • 13.3: Conditional probability (3)
    • 13.4: The general multiplication rule (3)
    • 13.5: Independence again (1)
    • 13.6: Tree diagrams (4)
    • 13.7: Bayes' rule
    • 13: Check Your Skills (10)
    • 13: Exercises (30)
    • 13: Exploring the Web
    • 13: Extra Problems (15)
    • 14.1: The binomial setting and binomial distributions
    • 14.2: Binomial distributions in statistical sampling (4)
    • 14.3: Binomial probabilities
    • 14.4: Using technology (3)
    • 14.5: Binomial mean and standard deviation (2)
    • 14.6: The Normal approximation to binomial distributions (3)
    • 14: Check Your Skills (9)
    • 14: Exercises (22)
    • 14: Exploring the Web
    • 14: Extra Problems (8)
    • 15.1: Parameters and statistics (2)
    • 15.2: Statistical estimation and the law of large numbers
    • 15.3: Sampling distributions (2)
    • 15.4: The sampling distribution of (3)
    • 15.5: The central limit theorem (2)
    • 15.6: Sampling distributions and statistical significance
    • 15: Check Your Skills (7)
    • 15: Exercises (19)
    • 15: Exploring the Web
    • 15: Extra Problems (16)
    • 16.1: The reasoning of statistical estimation (2)
    • 16.2: Margin of error and confidence level (1)
    • 16.3: Confidence intervals for a population mean (3)
    • 16.4: How confidence intervals behave (3)
    • 16: Check Your Skills (8)
    • 16: Exercises (7)
    • 16: Exploring the Web
    • 16: Extra Problems (8)
    • 17.1: The reasoning of tests of significance (1)
    • 17.2: Stating hypotheses (5)
    • 17.3: P-value and statistical significance (3)
    • 17.4: Tests for a population mean (2)
    • 17.5: Significance from a table (3)
    • 17.6: Resampling: significance from a simulation
    • 17: Check Your Skills (9)
    • 17: Exercises (13)
    • 17: Exploring the Web
    • 17: Extra Problems (8)
    • 18.1: Conditions for inference in practice (3)
    • 18.2: Cautions about confidence intervals (4)
    • 18.3: Cautions about significance tests (4)
    • 18.4: Planning studies: sample size for confidence intervals (2)
    • 18.5: Planning studies: the power of a statistical test (4)
    • 18: Check Your Skills (9)
    • 18: Exercises (25)
    • 18: Exploring the Web
    • 18: Extra Problems (11)
    • 19: Test Yourself (38)
    • 19: Supplementary Exercises (16)
    • 19: Extra Problems
    • 20.1: Conditions for inference about a mean (2)
    • 20.2: The t distributions (2)
    • 20.3: The one-sample t confidence interval (2)
    • 20.4: The one-sample t test (2)
    • 20.5: Using technology
    • 20.6: Matched pairs t procedures (1)
    • 20.7: Robustness of t procedures (2)
    • 20.8: Resampling and standard errors
    • 20: Check Your Skills (9)
    • 20: Exercises (18)
    • 20: Exploring the Web
    • 20: Extra Problems (12)
    • 21.1: Two-sample problems (1)
    • 21.2: Comparing two population means
    • 21.3: Two-sample t procedures (1)
    • 21.4: Using technology (1)
    • 21.5: Robustness again (4)
    • 21.6: Details of the t approximation (1)
    • 21.7: Avoid the pooled two-sample t procedures
    • 21.8: Avoid inference about standard deviations
    • 21.9: Permutation tests
    • 21: Check Your Skills (9)
    • 21: Exercises (23)
    • 21: Exploring the Web
    • 21: Extra Problems (8)
    • 22.1: The sample proportion (2)
    • 22.2: Large-sample confidence intervals for a proportion (2)
    • 22.3: Choosing the sample size (2)
    • 22.4: Significance tests for a proportion (3)
    • 22.5: Plus four confidence intervals for a proportion (3)
    • 22: Check Your Skills (7)
    • 22: Exercises (6)
    • 22: Exploring the Web
    • 22: Extra Problems (17)
    • 23.1: Two-sample problems: proportions
    • 23.2: The sampling distribution of a difference between proportions
    • 23.3: Large-sample confidence intervals for comparing proportions
    • 23.4: Using technology
    • 23.5: Significance tests for comparing proportions (3)
    • 23.6: Plus four confidence intervals for comparing proportions (1)
    • 23: Check Your Skills (6)
    • 23: Exercises (16)
    • 23: Exploring the Web
    • 23: Extra Problems (7)
    • 24: Test Yourself (24)
    • 24: Supplementary Exercises (15)
    • 24: Extra Problems (14)
    • 25.1: Two-way tables (1)
    • 25.2: The problem of multiple comparisons (1)
    • 25.3: Expected counts in two-way tables (1)
    • 25.4: The chi-square test statistic
    • 25.5: Cell counts required for the chi-square test
    • 25.6: Using technology (2)
    • 25.7: Uses of the chi-square test: independence and homogeneity (1)
    • 25.8: The chi-square distributions (1)
    • 25.9: The chi-square test for goodness of fit (5)
    • 25: Check Your Skills (10)
    • 25: Exercises (11)
    • 25: Exploring the Web
    • 25: Extra Problems (11)
    • 26.1: Conditions for regression inference
    • 26.2: Estimating the parameters (1)
    • 26.3: Using technology (2)
    • 26.4: Testing the hypothesis of no linear relationship (3)
    • 26.5: Testing lack of correlation (2)
    • 26.6: Confidence intervals for the regression slope (3)
    • 26.7: Inference about prediction (2)
    • 26.8: Checking the conditions for inference (2)
    • 26: Check Your Skills (9)
    • 26: Exercises (23)
    • 26: Exploring the Web
    • 26: Extra Problems (13)
    • 27.1: Comparing several means
    • 27.2: The analysis of variance F test (1)
    • 27.3: Using technology (1)
    • 27.4: The idea of analysis of variance
    • 27.5: Conditions for ANOVA (2)
    • 27.6: F distributions and degrees of freedom (2)
    • 27.7: Some details of ANOVA (3)
    • 27: Check Your Skills (2)
    • 27: Exercises (11)
    • 27: Exploring the Web
    • 27: Extra Problems (20)
    • 28.1: Comparing two samples: the Wilcoxon rank sum test
    • 28.2: The Normal approximation for W
    • 28.3: Using technology
    • 28.4: What hypotheses does Wilcoxon test?
    • 28.5: Dealing with ties in rank tests (2)
    • 28.6: Matched pairs: the Wilcoxon signed rank test
    • 28.7: The Normal approximation for W + (3)
    • 28.8: Dealing with ties in the signed rank test
    • 28.9: Comparing several samples: the Kruskal&ndashWallis test
    • 28.10: Hypotheses and conditions for the Kruskal&ndashWallis test
    • 28.11: The Kruskal&ndashWallis test statistic
    • 28: Check Your Skills (8)
    • 28: Exercises
    • 28: Exploring the Web
    • 28: Extra Problems
    • 29.1: Parallel regression lines (2)
    • 29.2: Estimating parameters
    • 29.3: Using technology
    • 29.4: Inference for multiple regression
    • 29.5: Interaction (4)
    • 29.6: The general multiple linear regression model (1)
    • 29.7: The woes of regression coefficients
    • 29.8: A case study for multiple regression
    • 29.9: Inference for regression parameters (1)
    • 29.10: Checking the conditions for inference (2)
    • 29: Check Your Skills (10)
    • 29: Exercises
    • 29: Exploring the Web
    • 29: Extra Problems
    • 30.1: Beyond one-way ANOVA
    • 30.2: Follow-up analysis: Tukey pairwise multiple comparisons
    • 30.3: Follow-up analysis: contrasts
    • 30.4: Two-way ANOVA: conditions, main effects, and interaction
    • 30.5: Inference for two-way ANOVA
    • 30.6: Some details of two-way ANOVA
    • 30: Check Your Skills
    • 30: Exercises
    • 30: Exploring the Web
    • 30: Extra Problems
    • 31.1: Processes
    • 31.2: Describing processes (2)
    • 31.3: The idea of statistical process control (1)
    • 31.4: charts for process monitoring (1)
    • 31.5: s charts for process monitoring
    • 31.6: Using control charts (1)
    • 31.7: Setting up control charts
    • 31.8: Comments on statistical control (2)
    • 31.9: Don't confuse control with capability
    • 31.10: Control charts for sample proportions
    • 31.11: Control limits for p charts (4)
    • 31: Check Your Skills (4)
    • 31: Exercises (3)
    • 31: Exploring the Web
    • 31: Extra Problems

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    14.2 Bernoulli’s Solution

    The St. Petersburg game was invented by the mathematician Nicolaus Bernoulli in the (18) th Century. He discussed it with his cousin Daniel Bernoulli, who published a famous solution in the St. Petersburg Academy Proceedings. (Hence the name of the game.)

    His solution: replace monetary value with real value, utility in other words. Recall that the more money you have, the less value additional money brings. So even though the payoffs in the St. Petersburg game double from ($2) to ($4) to ($8) etc., the real value doesn’t double. It grows more slowly.

    Figure 14.3: Bernoulli’s logarithmic utility function

    What is the real value of money then? How much utility does a gain of ($x) bring? Bernoulli proposed that utility increases “logarithmically” with money. In terms of dollars, (u($x)=log(x)) . Look at Figure 14.3 to see what this looks like.

    Notice how, for example, ($16) doesn’t have twice the utility of ($8) . In fact you have to go all the way up to ($64) to get twice the utility of ($8) . That’s because (64 = 8^2) , and logarithms have the property that (log(a^b) = b imes log(a)) .

    The difference this makes to the St. Petersburg game is displayed in Figure 14.4. The rectangles don’t all have the same area of (1) anymore, they get smaller instead. In fact, Bernoulli showed that the total area of all the rectangles is only about (1.39) . In other words, the expected utility of the St. Petersburg game is about (1.39) , the equivalent of a ($4) gain (because (log(4) approx 1.39) ).

    Figure 14.4: Dollars vs. utils in the St. Petersburg game

    So if Bernoulli is right about the utility of money, the fair price for the St. Petersburg game is only about ($4) . And in fact that’s about what most people say they would pay.


    Introduction to Probability - MATH/STATS 425, Winter 2012

    Class meets: Section 3 MWF 1:10 - 2:00 in 1068 East Hall Section 7 MWF 2:10 - 3:00 in 1084 East Hall.

    Office Hours: Tu 3:30-5:00 pm, W 10:30 - 12:00 am, F 10:00 - 11:00 am in 4844 East Hall.

    Prerequisites: This course makes serious use of the material from the Calculus sequence (Math 215, 255 or 285). You should have solid knowledge of differentiation and integration, in particular of the geometric ideas behind these concepts (tangent lines, local extrema, areas, volumes). This course requires a working knowledge of multivariate differentiation and integration (including change of variables). If you do not have a strong background in Calculus, please do not take this course we do not want you to fail because of your inadequate background.

    Previous exposure to basic combinatorics (combinations and permutations) is generally expected. If you do not have this background and wish to take this course, please carefully study Sections 1.1-1.4 of the textbook and the self-test problems at the end of Chapter 1.

    Course Description: Basic concepts of probability are introduced, and applications to other sciences are noted. The emphasis is on concepts, calculations, derivations and problem-solving. Topics include methods of both discrete and continuous probability, conditional probability, independence, random variables, joint distributions, expectations, variances, covariances, and limit laws.

    Text: Sheldon Ross, A first course in probability. Pearson Prentice Hall, 8th ed. (2010). ISBN: 9780136033134.
    The course covers most of Chapters 1--7 and a part of Chapter 8.

    • Homework (20%). Scroll down to see it. One lowest homework will be dropped. To form study groups and for online collaboration, feel free to use the chat room and forums on the Ctools class page.
    • Quizzes (20%). One lowest quiz will be dropped.
    • Midterm Exam (25%). Wednesday, February 22, in class. Covers Chapters 1 - 4.
    • Final Exam (35%). Both sections: Friday, April 20, 1:30-3:30 pm, in 1400 CHEM. Covers the whole course.

    Missing/late work: There will be no make-up for the quizzes or exams for any reason. A missed exam counts as zero points, with the following exception. If you miss a midterm exam due to a documented medical or family emergency, the exam's weight will be added to the final exam. Late homework will not be accepted. In case of a medical or road emergency that prevents you from attending the class, you can e-mail me your scanned or typed homework as a single pdf file, along with short explanation of what happened, on the day when the homework is collected by 5 pm.

    Course Schedule: It is a useful practice to read ahead the sections to be covered. To see what material is coming next, have a look at my previous Math/Stats 425 course. After each class, the schedule will be updated with the description of the material we covered in class, examples and self-test problems, and new homework. It is a useful practice to work on the examples and self-test problems. Their solutions are included in the back of the textbook do not turn them in as a part of homework.

    Wednesday, January 4 Multiplication Principle (a.k.a. the basic principle of counting) (1.1). Permutations (1.3). Combinations (1.4 begin). Lecture notes All examples in Sections 1.2 Examples 3a, 3b, 3c 4a, 4b, 4c. Self-test problems (p.20-21): 1, 2, 3, 4, 6. Homework 1, due January 11 (p. 16-17): 3, 5, 7(a,b), 15, 17, 21. Friday, January 6 Combinations contd. (1.4). Binomial theorem. Multinomial coefficients (1.5). Lecture notes Examples 4d, 4e 5a, 5b, 5c. Self-test problems (p. 20-21):5, 8, 9, 10, 11, 12. Homework 1, due January 11 (p. 16-17): 9, 13, 28. Monday, January 9 Sample space and events. Operatoins on events (2.1-2.2). Lecture notes All Examples in 2.1-2.2. Self-test problems (p. 56-57): 1. Homework 2, due January 18 (p. 50-54): 1, 3, 6. Wednesday, January 11 Quiz 1 Axioms of probability (2.3). Properties of probability. Inclusion-Exclusion Principle (2.4 up to Example 4a). Lecture notes All examples in those sections. Self-test problems (p. 56-57): 2, 4, 14, 15. Homework 2, due January 18 (p. 50-54): 9, 11, 12 (hint: use Venn diagram for this last problem). (Do not do Problem 14 at this time it will be assigned later in Homework 3.) --> Friday, January 13 Inclusion-Exclusion Principle continued. Sample spaces having equally likely ourcomes (2.5 begins). Lecture notes Examples 5a-d, recall the Birthday Problem (Example 5i). Self-test problems (p. 56-57): 5, 6, 7, 8. Homework 2, due January 18 (p. 50-54): 8, 17, 21. Monday, January 16: no class (Martin Luther King Jr. Day) Wednesday, January 18 Quiz 2 Generalized Inclusion-Exclusion Principle (Proposition 4.4). The Matching Problem (Example 5m). More problems on equally likely outcomes. Lecture notes Examples 5e, l, n. Self-test problems (p.56-57): 10, 12, 13, 17, 18, 20. Homework 3, due January 25 (p. 50-54): 14, 23, 25, 28, 32. Friday, January 20 Conditional Probabilities. The Law of Total Probability (3.1-3.2). Lecture notes Examples 2b, e, f, the first example on p.21 of the lecture notes. Self-test problems (p.114-116): 2, 3, 5, 9a. Homework 3, due January 25 (p.102-110): 16, 19b, 21. Monday, January 23 Bayes Formula (3.3). Lecture notes Examples 3a, c, d (covered in class), e, f, k (covered in class), n. Homework 4, due February 1 (p.102-110): 15, 18, 19a, 26, 35, 90 (pass to the complement, use the inclusion-exclusion principle). Wednesday, January 25 Quiz 3 Independent events (3.4). Simple random walk. Lecture notes Examples 4b, c, g, h, i. Self-test problems (p.114-116): 4, 9b, 10, 11, 12, 15, 16 (should refer to problem 3.66b), 21, 23. Homework 4, due February 1 (p.102-110): 50, 57(a,b), 64, 66(a) (in some editions there is a typo -- this problem should refer to Figure 3.4). Friday, January 27 Random variables (4.1). Discrete random variables (4.2). Probability mass function (pmf) and cumulative distribution function (cdf). Lecture notes Examples 4a, b, c, d. No additional homework. Monday, January 30 Expected value (4.3). Examples: lottery group testing. Lecture notes Examples 3a, b, c, d. Self-test problems (p.183-185): 1, 2, 3, 4 (if you dare), 6. Homework 5, due February 8 (p.172-179): 1, 13, 17, 19. Note: the textbook often calls CDF "the distribution function". Wednesday, February 1 Quiz 4 Expected value of a function of a random variable (4.4). Variance (4.5). Lecture notes Examples 4a, 5a. Self-test problems (p.183-185): 5. Homework 5, due February 8 (p.172-179): 21, 25, 30, 35, 37. Friday, February 3 Example: the matching problem (expectation and variance). Bernoulli and binomial random variables (4.6). Lecture notes Examples 6a, b, c, e, f. Self-test problems (p.183-185): 9, 10, 11a, 12, 13. Homework 5, due February 8 (p.172-179): 42, 43, 48. Monday, February 6 Poisson distribution (4.7). Example: People vs. Collins trial. Lecture notes Examples 7a, b, c. Self-test problems (p.183-185): 14, 15. More examples are added for Feb.3 class above Homework 6, due February 15 (p.172-179): 45, 52, 53, 54, 59. Wednesday, February 8 Quiz 5 Binomial and Poisson distributions (continued). Example: jury duty letters. Lecture notes Self-test problems (p.384-387): 3, 4. Homework 6, due February 15 (p.172-179): 64(a,b) (p.373-379): 7 (express the number of chosen objects as a sum of 10 indicators), 8. Friday, February 10 Computing expectations by conditioning (from 7.5). Geometric distribution (4.8.1). Lecture notes Examples: from Section 7.5: 5c, h from section 4.8: 8a, b. Self-test problems (p.384-387): 17 (p.183-185): 22. Homework 6, due February 15 (p.373-379): 48(a,b), 53, 58 (condition on the outcome of the first flip). Monday, February 13 Continuous distributions (5.1). Uniform distribution (5.3). Lecture notes Examples: 1a, b, c (all very useful), 3c. Self-test problems (p.229-213): 1, 2, 7. Homework 7, due February 22 (p.224-227): 1, 4, 11, 13. Wednesday, February 15 Quiz 6 Transformations of random variables. Lecture notes Example from Section 5.2: 1d. Homework 7, due February 22: additional problems. Friday, February 17 Expectation of continuous random variables (5.2). Expectation and variance of the uniform distribution (Ex.3a). Lecture notes Self-test problems (p.229-231): 3, 4, 5, 6. Homework 7, due February 22 (p.224-227): 7, 14. Monday, February 20 Standard normal distribution (5.4). Lecture notes Wednesday, February 22 Midterm Exam. In class. Covers Chapters 1 - 4. Sample problems: Practice Exam (due to Prof. Montgomery): problems 1, 5. Solutions. More exam problems (due to Prof. Montgomery): try problems 1(a), 2, 3, 5, 6, 9, 10, 12. Practice Exam (due to Prof. Derksen): problems 1, 2, 4, 5. Practice Exam (due to Prof. Montgomery): problems 3(except b), 4(except d,f). Solutions. Practice Exam (due to Prof. Khoury): problems 1, 5, 8, 9, 11. Practice Exam (due to Prof. Khoury): problems 3, 4a, 5, 9, 10, 11. Practice Exam (due to Prof. Khoury): problems 2, 3, 7. Practice Exam (due to Prof. Khoury): problems 2, 8(a,b). Practice Exam (due to Prof. Khoury): problems 1, 6, 9. Practice Exam (due to Prof. Khoury): problems 1, 3, 8. Homework 1: all problems, Homework 2: 9, 11, 8, 21 Homework 3: 14, 28, 32, 16, 19, 21 Homework 4: 15, 35, 57, 66a Homework 5: 17, 19, 43, 37 Homework 6: all problems. Friday, February 24 Normal distribution with general mean and varaince (5.4). Lecture notes Examples: 4b, c, d, e. Self-test problems (p.229-231): 8, 9, 10, 11. Homework 8, due March 7 (p.224-227): 15, 16, 17, 18, 22. Winter break: February 27 - March 2 Monday, March 5 Normal approximation to the binomial distribution (5.4.1). Lecture notes Examples 4f, g, h, i. Self-test problems (p.229-231): 12, 17. Homework 9, due March 14 (p.224-227): 20, 27, 28. Wednesday, March 7 Quiz 7 Exponential distribution (5.5). Lecture notes Examples 5a, b, c, d. Self-test problems (p.229-231): 18, 19. Homework 9, due March 14 (p.224-227): 31, 32, 34, 39. Friday, March 9 Joint distributions (6.1). Lecture notes Examples 1a, b, c, d, e. Self-test problems (p.293-296): 6.2, 6.3, 6.6(a,c), 6.7(b,c,d,e,f). Homework 9, due March 14 (p.287-291):6, 8, 9(a,b,c,e), 10. Monday, March 12 Independent random variabes (6.2). Lecture notes Examples 2a, d, f. Homework 10, due March 21 (p.287-291): 15, 16, 20, 22, 23, 27. Wednesday, March 14 Quiz 8 Sums of independent random variables (6.3). Lecture notes No additional homework problems today. Friday, March 16 Sums of independent random variables (6.3). Convolutions. Sums of independent normal random variables. Lecture notes Homework 10, due March 21 (p.287-291): 29, 30, 31. Monday, March 19 Sums of independent random variables (Bimonial, Poisson, exponential). Gamma distribution (6.3 continued). Lecture notes Homework 11, due March 28 (p.287-291): 28, 33(a,b). Wednesday, March 21 Quiz 9 Conditional distributions (6.4-6.5). Lecture notes Homework 11, due March 28 (p.287-291): 40, 41, 42 (compute the conditional pdf of Y given X=x). Friday, March 23 Conditional distributions (6.4-6.5 continued). Lecture notes Homework 11, due March 28 (p.287-291): 48, and an additional problem. Monday, March 26 Transformations of joint distributions (6.7). Lecture notes Homework 12, due April 4 (p.287-291): 52, 54, 55, 56(a). Wednesday, March 28 Quiz 10 Properties of expectation (7.1). Expectation of sums of random variables (7.2-7.3). Lecture notes Homework 12, due April 4 (p.373-379): 4, 5, 11, 21. Friday, March 30 Coupon collector's problem (Section 7, Examples 3d, 2i). Expectation of product of independent random variables (Proposition 4.1). Lecture notes Homework 12, due April 4 (p.373-379): 19(a). Monday, April 2 Covariance, correlation (7.3). Lecture notes Homework 13, due April 11 (p.373-379): 50, 38, 40. Additional problem. Wednesday, April 4 Quiz 11 Properties of covariance. Variance of a sum. Lecture notes Homework 13, due April 11 (p.373-379): 6, 31, 36, 37, 42, 45, 63(a). Friday, April 6 Applications to statistics: sample mean and sample variance (Ex. 4a). Lecture notes Homework 13, due April 11 (p.373-379): 65. Monday, April 9 Moment generating functions (7.7). Lecture notes Wednesday, April 11 Quiz 12 Central Limit Theorem (8.3). Lecture notes Homework 14, will not be collected (p.413-414): 5, 7, 8, 14. Friday, April 13 Markov's and Chebyshev's inequalities. Weak law of large numbers. Monday, April 16 Review. Practice Exam 0 Practice Exam 1 Practice Exam 2 Practice Exam 3 Homework 7: 4, 7, 13, 14, additional problems 1, 2. Homework 8: 16, 22. Homework 9: 28, 32, 8, 9, 10. Homework 10: 16, 20, 22, 27, 31. Homework 11: 28, 41, 42, 48. Homework 12: 54, 56(a), 4, 5, 11, 21. Homework 13: 50, 31, 36, 37, 42, 63(a). Homework 14: 5, 7, 8, 14.
    This formula sheet will be available during the exam. In addition, you may bring two index cards (3x5 in) to the exam. You may write whatever you like on one side of each card. Please only write by hand on the cards photocopyied or printed cards are not allowed. Friday, April 20 Final Exam. 1:30-3:30 pm, in 1400 CHEM. Covers the whole course.

    Course webpage: http://www-personal.umich.edu/

    romanv/teaching/2011-12/425/425.html
    Also see the Ctools class page which contains the archive of e-mail messages to the class, forums and chat room.


    Watch the video: Probability: Binomial Theorem (May 2022).