Geometric Distribution A probability distribution to determine the probability that success will occur on the nth trial of a binomial experiement

Geometric Distribution Repeated binomial trials Continue until first success Find probability that first success comes on nth trial Probability of success on each trial = p

Geometric Probability

A sharpshooter normally hits the target 70% of the time. Find the probability that her first hit is on the second shot. Find the mean and the standard deviation of this geometric distribution.

A sharpshooter normally hits the target 70% of the time. Find the probability that her first hit is on the second shot. P(2)=p(1-p) n-1 = .7(.3)2-1 = 0.21 Find the mean  = 1/p = 1/.7 1.43 Find the standard deviation

Poisson Distribution A probability distribution where the number of trials gets larger and larger while the probability of success gets smaller and smaller

Poisson Distribution Two outcomes : success and failure Outcomes must be independent Compute probability of r occurrences in a given time, space, volume or other interval  (Greek letter lambda) represents mean number of successes over time, space, area

Poisson Distribution

The mean number of people arriving per hour at a shopping center is 18. Find the probability that the number of customers arriving in an hour is 20. r = 20  = 18 Find P(20) e = 2.7183

The mean number of people arriving per hour at a shopping center is 18.

Poisson Probability Distribution Table Table 4 in Appendix II provides the probability of a specified value of r for selected values of .

Using the Poisson Table  = 18, find P(20):

Poisson Approximation to the Binomial Distribution The Poisson distribution can be used as a probability distribution for “rare” events.

“Rare” Event The number of trials (n) is large and the probability of success (p) is small.

If n  100 and np < 10, then The distribution of r (the number of successes) has a binomial distribution which is approximated by a Poisson distribution . The mean  = np.

Use the Poisson distribution to approximate the binomial distribution: Find the probability of at most 3 successes.

Using the Poisson to approximate the binomial distribution for n = 240 and p = 0.02 Note that n  100 and np = 4.8 < 10, so the Poisson distribution can be used to approximate the binomial distribution. Find the probability of at most 3 successes: Since  = np = 4.8, we use Table 4 to find P( r  3) =.0082 + .0395 + . 0948 + . 1517 = .2942