Integrated Learning System

Week 3, Lecture 1

More Discrete Probability

Distributions

Objectives:

At the end of this presentation you should be able to find probabilities using the geometric distribution and the Poisson distribution.

Definition

A geometric distribution is a discrete probability distribution of a random variable x that satisfies the following conditions:

1. The trial is repeated until a success occurs.

2. The repeated trials are independent.

3. The probability of success p is constant for each trial.

Geometric Probability

Function

P(x) = pqx-1 where q = 1 – p.

Geometric Probability

Example

A cereal maker places a game piece in its cereal boxes. The probability of winning a prize in the game is ¼. Find the probability that you:

1. Win your first prize with your 4th purchase.

2. Win your first prize with your 1st, 2nd, or 3rd purchase. 3. You do not win a prize with your first four purchases. (Problem 10, page 193, Larson and Farber)

Geometric Distribution

So, what does the distribution we have just used look like?

Geometric Distribution for p = 0.25

0.3

0.25

P (x )

0.2

0.15

0.1

0.05

0

1

2

3

4

5

6 x 7

8

9

10

Using Technology: TI-83 geometpdf(p,x) – Probability density function geometcdf(p,x) – Cumulative probability density function Using Technology: EXCEL

While EXCEL contains many of the probability density functions that we will use in this course, it does not have a geometric probability density function. However, the geometric probability density function is fairly simple and it is easy to enter as a cell function.

Exercise

Basketball Player Shaquille O’Neal makes a free throw shot about 53.4% of the time. Find the probability that:

1. The first shot O’Neal makes is the second shot.

2. The first shot O’Neal makes is the first or second shot.

3. O’Neal does not make two shots.

Definition

The Poisson distribution is a discrete probability distribution that satisfies the following

1. The experiment consists of counting the number of times x that an event occurs in a given interval.

2. The probability of the event is the same for each interval.

3. The number of occurrences is independent from interval to interval.

Poisson Probability

Function

xe

x e

P x

, in sometextswrittenP x x! x!

In the versions of the function above and represent the mean arrival rate or rate of occurrence for the event of interest.

is the lower case Greek letter