Bernoulli

1
Bernoulli

• Bernoulli population is a population in which
  each element is one of two possibilities. The two
  possibilities are usually designed as success and
  failure.
• The probability of success is p which means that
  the probability of failure
• is 1-p.
• A Bernoulli trial is observing one element in a
  Bernoulli population.

2
Example

• Ex1. The toss of a coin results in a Bernoulli
  population.
• Ex2. The homes in Dallas either heat with gas or
  not. Choices to heat with gas or not is a
  Bernoulli population.
• Ex3. In a local election, the voters either
  favor a candidate for mayor or do not. The
  choices of voters result in a Bernoulli
  population.

3
Binomial experiment

• A Binomial experiment is an experiment that
  consists of n repeated independent Bernoulli
  trials in which the probability of success on
  each trial is ? and the probability of failure
  on each trial is 1- ? .

4
Criteria for a Binomial Setting

• 1.consists of n Bernoulli trials (each trial
  yields either S or F.
• 2.For each trial, P(S) ? ,and P(F)1- ? .
• 3.The trials are independent.
• Ex1. The toss of a coin 3 times

5
binomial random variable.

• The random variable x, which gives the number of
  success in the n trials of Bernoulli experiment,
  is called a binomial random variable. The sample
  space of x is
• Sx 0, 1, 2, , n.
• Ex1. The number of head after tossing a coin 10
  times.

• The binomial random variable is a discrete random
  variable.

6
Question Are the following random variables
binomial random variable?

• The number of getting head after flipping a coin
  10 times.
• 40 of all airline pilot are over 40 years of
  age. The number of pilots who are over 40 out of
  15 randomly chosen pilots.
• Suppose a salesperson makes sales to 20 of her
  customers, the number of customers until her
  first sale.
• A room contains 6 women and four men, Three
  people are selected to form a committee. The
  number of women on the committee.

7

• Suppose a couple plans to have 3 children. The
  chance they have a boy is 0.2. The gender of one
  child is independent of the gender of another
  child.
  Let X be the number of boys they have.

8

• If the chance a child is a boy is 0.2, whats the
  chance a child is a girl?
• How many gender sequences (i.e. BBB, BBG, BGG,
  etc) are possible?

9
We want to fill in the probability distributionn
below
P(X 0) P(all three are girls) P(GGG)
(0.8)3
P(X 3) P(all three are boys) P(BBB) (0.2)3
10

• P(X 1)
• P(only one boy)
• P(GGB) P(GBG) P(BGG)
• (0.8)2(0.2) (0.8)2(0.2)
• (0.8)2(0.2)
• 3 (0.8)2(0.2)

11

• Similarly, P(X2) equals 3(0.2)2(0.8). Now we
• can complete the probability distribution of X.

12
Binomial distribution
Probability of k successes out of n trials is
given by
where
13

• Ex 50 of homes in Dallas are heated with gas.
  Let Y of homes out of 6 that are heated with
  gas.
• The sample space of the value of y is
• Sy 0, 1, 2, 3, 4, 5, 6
• It is a Binomial random variable. What is the
  distribution of Y?

14
Solution

• When n is very large, computing the probabilities
  becomes very tedious. In fact, We can use
  computer or binomial table to find out the
  probabilities for certain values of n, k and ? .
• Look at table B.1 on Appendix B.

15
How the two parameters (n, ?) affect binomial
distribution?

• Fix n, ? is more close to .5, the distribution
  is more near to a bell curve. If ? is more
  close to 0, the distribution is more skewed to
  the right. If ? is more close to 1, the
  distribution is more skewed to the left.

16
Fix n, let ? change
17
Fix ? , let n change

• When the success probability ? on each trial is
  fixed and n is lager, the distribution is more
  close to a bell curve

18
Fix ? , let n change
19
mean and standard deviation

• For binomial distribution
• Meanµ n ?
• Variances 2 n? (1- ? )
• Standard deviation s

20
Example 4.25(page 274)

• Suppose that the probability of successfully
  rehabilitating a convicted criminal in a penal
  institution is .4. Let r represent the number
  successfully rehabilitated out of a random sample
  of ten convicted criminal, find out the mean and
  the standard deviation of the variable r.

21
Example 4.13

• 30 of homes in Dallas are heated with gas. Let
  Y of homes out of 6 that are heated with gas.
• What is the probability of Y

22

• 0 0.117649
• 1 0.302526
• 2 0.324135
• P(Y

23
Probability of event in the Binomial case

• Calculate the unknown probability of random
  variable y with n4 and ?.2
• P(y2)
• P(y
• P(y2)
• P(y3)