The Geometric and Poisson Distributions

1. The Geometric and Poisson Distributions

2. Geometric Distribution – A geometric distribution shows the number of trials needed until a success is achieved. • Example: When shooting baskets, what is the probability that the first time you make the basket will be the fourth time you shoot the ball?

3. Conditions for Geometric Distribution 1. A trial is repeated until a successoccurs • The repeated trials are independent for • each trial 3. The probability of success p is constant • The random variable x represents the number of the trial in which the first success occurs

4. Geometric Distribution Equation • P (x) = p(q)x – 1 p = the probability of success q = 1 – p x is the random variable(what you are looking for!)

5. Connection to Geometric Series • Think of this as a geometric series note the similarities to the distribution equation. • Remember in geometric series in algebra was the first term and r = the common ratio and n = the term you were looking for. • i.e. for n=4 we would find the 4th term in the series. In this case =24 The series would look like 3, 6, 12, 24 • The form for geometric distribution gives us the probability of the n term because we are using probabilities in the formula

6. Example Suppose that 40% of students who drive to school carry jumper cables. If your car has a dead battery, and you aren’t one of the prepared students, how many students will you have to ask before you find one with jumper cables? a) Find the probability that the 3rd person you ask is the first person who has jumper cables. Ans: The calculator steps are geometpdf(p,x) or geompdf(0.4,3)= 0.144 (pdf – for precise or exactly!) b) Find the probability that the cables are found on or before the fourth person. Ans: () = P(x = 4) + P(x = 3) + P(x = 2) + P(x = 1) =+++= 0.8704 Orgeometcdf(0.4,4) = 0.8704 (cdf – for cumulative adding everything down!)

7. Poisson Distribution – Use when you are looking for the number of times something occurs during a given interval. • Example: How many accidents will occur at the intersection of Chandler Blvd and Arizona Ave between 11:30 p.m. – 12:30 p.m.

8. Poisson Distribution • The experiment counts how many times, x, an event occurs in a given interval. • The probability of the event occurs is the same for each interval • The number of occurrences in one interval is independent of the number of occurrences in other intervals.

9. Poisson Equation • Where µ is the mean number of occurrences per interval unit • Random variable ‘x’ is what you are looking for • e is an irrational number approximately equal to 2.71828

10. Example for Poisson Distribution • If a hockey player is averaging 1.4 points per game, a) What is the probability that they will have 2 points in a game? Ans: Using the calculator: Poissonpdf( µ, x) Or poissonpdf(1.4, 2) = 0.242 • What is the probability that more than 2 points in a game will occur? Ans: 1 – (+) = 1 – (0.242 + 0.345 + 0. 247)=0.167 Or 1 - Poissoncdf(1.4, 2) = 0.167

11. Complete the 4.3 Classwork