Negative binomial distribution

Negative binomial distribution

1. NADEEM UDDIN ASSOCIATE PROFESSOR OF STATISTICS NEGATIVE BINOMIAL DISTRIBUTION

2. Negative Binomial Experiment A negative binomial experiment is a statistical experiment that has the following properties : 1-The experiment consists of x repeated trials . 2-Each trial can result in just two possible outcomes. We call one of these outcomes a success and the other, a failure. 3-The probability of success, denoted by P, is the same on every trial . 4-The trials are independent; that is, the outcome on one trial does not affect the outcome on other trials . 5-The experiment continues until k successes are observed, where  k is specified in advance.

3. Consider the following statistical experiment. You flip a coin repeatedly and count the number of times the coin lands on heads. You continue flipping the coin until it has landed 3 times on heads. This is a negative binomial experiment because: 1-The experiment consists of repeated trials. We flip a coin repeatedly until it has landed 3 times on heads. 2-Each trial can result in just two possible outcomes – heads or tails. 3-The probability of success is constant ( 0.5) on every trial. 4-The trials are independent; that is, getting heads on one trial does not affect whether we get heads on other trials. 5-The experiment continues until a fixed number of successes have occurred; in this case, 3 heads.

4. Notation   The following notation is helpful, when we talk about negative binomial probability . x: The number of trials required to produce  k  successes in a negative binomial experiment . k: The number of successes in the negative binomial experiment. P: The probability of success on an individual trial . q : The probability of failure on an individual trial.

5. Negative Binomial Random Variable The number X of trials to produce k successes in a negative binomial experiment is called a negative binomial random variable . Negative Binomial Distribution A negative binomial random variable is the number X of repeated trials to produce k successes in a negative binomial experiment. The probability distribution of a negative binomial random variable is called a negative binomial distribution. The negative binomial distribution is also known as the Pascal distribution .  

6. Example-1   You are surveying people exiting from a polling booth and asking them if they voted independent. The probability that a person voted independent is 25%. What is the probability that 15 people must be asked before you can find 5 people who voted independent? Solution K = 5, x = 15, p = 0.25, q = 0.75  

7. Example-2 Suppose that the probability is 0.8 that any given person will believe a tale about life after death. What is the probability that the sixth person to hear this tale is the fourth one to believe it ?      Solution K = 4, x = 6, p = 0.8, q = 0.2  

8. Example-3 A football player, his success rate of goal hitting is 70%. What is the probability that player hits his third goal on his fifth attempt?   Solution K = 3, x = 5, p = 0.7, q = 0.3    

9. Example-4 You draw cards from a deck (with replacement) until you get four kings. What is the probability that you will draw exactly 20 times   Solution K = 4, x = 20, , q = 0.923    

10. Example-5 Find the probability that a person tossing 3coins will get either all heads or all tails for the second time on the fifth toss Solution K = 2, x = 5, , q = 0.75