Geometric DistributionChapter 8 Day 5

What is Geometric Distribution?

X is a ________________ ______________ ______________if it follows the conditions of a geometric setting:

(1) There are only _______ outcomes: _________ or ___________.

(2) The probability of success p is the ____________ for each observation. (Probability is _____________)

(3) The n observations are ________________.

(4) Trials ____________ until you get a ______________

Some facts:

•1) Because n is ________ fixed there could be an ____________ ______________of x values.

•2) The Probability histogram for a geometric distribution is ALWAYS _____________ _____ _____ ___________.

Geometric Formulas

•The probability that X is equal to n is

•This gives the probability that the _______ __________will occur on trial X.

Example:•There is a 90% chance that it will rain one day this week. Find the probability that:•A) The first rain storm occurs on the fourth day.

•B) It rains exactly 2 days.

•C) It rains at least three days.

•D) The first day of rain is on the last day of the week.

Example:

•We are going to flip a coin until a head appears. What is the probability that the first head will appear on the fifth or sixth toss?

Another Geometric Formula

•The probability that it takes more than n trials to see the first success is

•Example: Find the probability that it takes more that 6 rolls to observe a 3.

Using the calculator

•For exactly n number of rolls, P(x = n):

• use

•For less than or equal to n number of rolls, :

• use

•For more than n number of rolls,

• use

( )P x n

( ) :P x n

Mean and Variance of Geometric Distibutions

•The mean or expected value of X is:

•

•The Variance of X is :

•

Example

•Find the expected value and standard deviation of observing a 3 when rolling a die.

Example:• There is a probability of 0.05 that a new Toyota will

experience certain engine problems. Suppose that a number of consumers buy a new Toyota. We are interested in the number of cars purchased until the first engine problem is observed.

• 1) Define the random variable of interest.

2) What is the distribution of X? Verify your answer.

3) Find the probability that exactly 10 consumers must buy a car in order to observe the first engine problem.

4) Construct a probability table for X (up to X = 5).

5) How many new cars purchased would you expect to have in order to observe the first engine problem?

6) What is the probability that the number of cars purchased before the first engine problem is observed is at most 7?

Example:

•Suppose X has a geometric distribution with p = 0.7

•Graph P(X) for X = 1, 2, 3, 4.

•Compute µ and σ.

•The most common data collection method in consumer and market research is the telephone survey. A major problem with consumer telephone surveys, however, is nonresponse. How likely are consumers to be at home to take the call and, if at home, how likely are they to take part in the survey? To answer these and other questions, R. A. Kerin and R. A. Peterson directed a study of over 250,000 randomdigits dialings of both listed and unlisted telephone numbers across the United States (Journal of Advertising Research, Apr./May 1983). The probability of a completed interview was assessed to be 0.084.

•a) What is the probability that five calls must be made before the first completed interview is obtained?

•b) What is the probability that at least five calls must be made before the first completed interview is obtained?

•c) Find the mean number of calls that must be made before the first completed interview is obtained? Find the standard deviation.

•A manufacturing process produces, on average, 3% defective items. The company ships 12 items each box and wishes to guarantee no more than 1 defective item per box. What is the probability that the box will fail to satisfy the guarantee?

•Suppose it is known that 80% of the people exposed to the flu virus will contract the flu. Out of a family of six exposed to the virus, what is the probability that

•A) No one will contract the flu?

•B) All will contact the flu?

•C) Exactly two will get the flu?

•D) At least two will get the flu?

•E) How many would you expect to get the flu? Also find the standard deviation.