Chapter 3 - Lecture 5 The Binomial Probability Distribution

Transcription

1 Chapter 3 - Lecture 5 The Binomial Probability October 12th, 2009

2 Experiment Examples Moments and moment generating function of a Binomial Random Variable

3 Outline Experiment Examples A binomial experiment is one that satisfies the four following requirements The experiment consists of a sequence of n smaller experiment called trials, where n is fixed in advance of the experiment Each trial can result in one of the same two possible outcomes success or failure. The trials are independent, so that the outcome on any particular trial does not influence the outcome on any other trial The probability of success is constant from trial to trial.

4 Examples Outline Experiment Examples Toss of a coin Roll a die, with for example S = {1, 2} and F = {3, 4, 5, 6} Birth of a child

5 Experiment Examples Examples - a small issue Suppose in Stat 318 there are 50 students, 44 men and 6 women. I want to select a committee of three to communicate all the requests to the instructor. The first one is the President, the second one is the vice president and the third one is just a member. Suppose in Penn State there are students, with only being women. I want to select a committee of three to communicate all requests to the President of the University. The first one is the President, the second one is the vice president and the third one is just a member. Are the above experiments, binomial experiments? Are there any issues with them?

6 Rule of thumb Outline Experiment Examples If we sample without replacement from a large sample, we will consider that it is a binomial experiment if our sample size is less than 5% of the population of interest.

7 Outline Moments and moment generating function of a Binomial Random When we have a binomial experiment consisting of n trials, the binomial random variable X associated with this experiment is defined as the number of successes among the n trials

8 Moments and moment generating function of a Binomial Random Binomial Every random variable has a distribution A binomial random variable has the Binomial distribution which is affected by two parameters, the number of trials n and the probability of success. The distribution is denoted as B(n, p)

9 Example Outline Moments and moment generating function of a Binomial Random If X B(n, p) and we want to find the P(X = x) we use the following formula: ( ) n p x (1 p) n x, x = 0, 1,..., n P(X = x) = x 0, otherwise

10 Binomial Tables Outline Moments and moment generating function of a Binomial Random Imagine that you have a random variable X B(25, 0.2). How would you find P(X < 13)?

11 Expected value and Variance Moments and moment generating function of a Binomial Random If X B(n, p) then: E(X ) = np var(x ) = np(1 p) Example: If X B(7, 0.35) find the expected value the variance and the standard deviation of X.

12 Moment Generating Function Moments and moment generating function of a Binomial Random If X B(n, p) then M X (t) = (pe t + 1 p) n Proof? Find E(X ) and var(x ) of a using the moment generating function

13 Section 3.5 page , 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 75, 76, 77, 78, 79