Geometric Distributions Section 8.2

The 4 “commandments” of Geometric Distributions Only 2 outcomes for each trial: success or failure. Only 2 outcomes for each trial: success or failure. Same probability of success for each trial Same probability of success for each trial All independent trials. All independent trials. The count of interest is the # of trials required to get the first success. The count of interest is the # of trials required to get the first success. The geometric setting is somewhat similar to the binomial setting, the basic difference being that the geometric setting does not have a fixed number of observations.

The probabilities of a geometric sequence X1234 P(X)p(1-p)p (1-p) 2 p (1-p) 3 p Rule for calculating geometric probability If X has a geometric distribution with probability p of success and (1-p) of failure on each observation, the possible values of X are 1,2,3… If n is any of these values, then the probability that the first success occurs on the nth trial is… P( X=n) = (1-p) n-1 p

Geometric cont’d If X is a geometric random variable with probability of success p on each trial. The mean (expected value) of the random variable, the expected number of trials required to get the first success, is If X is a geometric random variable with probability of success p on each trial. The mean (expected value) of the random variable, the expected number of trials required to get the first success, is The variance is The variance is So the standard deviation is So the standard deviation is The probability that it takes more than n trials to see the first success is: The probability that it takes more than n trials to see the first success is: P(X>n) = (1-p) n

Calculator geometpdf(p,X) p being the probability of success and X being the number of the trial which the first success occurs. geometpdf(p,X) p being the probability of success and X being the number of the trial which the first success occurs. goemetcdf(p,X) where X is the maximum number of trials to get the first success goemetcdf(p,X) where X is the maximum number of trials to get the first success

Example The pitcher is starting a new inning. The probability that he’ll throw a strike is still.6. The pitcher is starting a new inning. The probability that he’ll throw a strike is still.6. What is the expected number of pitches needed to pitch a strike? What is the probability that he will throw his first strike on the 5 th pitch? What is the probability that it will take more than 5 pitches before he throws his first strike? 1/p≈ pitches geometpdf(0.6, 5)= = – geometcdf(0.6,5)=

Conclusion Binomial setting is for a fixed number of trials whereas the geometric setting has no fixed number of trials. Review the 4 commandments! Learn them, accept them, embrace them! There are only 2 outcomes, success or failure and they each have a fixed probability. What is the shape of a geometpdf probability histogram when the probability of success is 0.3? p=0.5? p=0.8? What is the shape of a geometcdf probability histogram? Right skewed Left skewed