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4.3 Geometric distribution  (Page 3/14)

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Assume that the probability of a defective computer component is 0.02. Components are randomly selected. Find the probability that the first defect is caused by the seventh component tested. How many components do you expect to test until one is found to be defective?

Let X = the number of computer components tested until the first defect is found.

X takes on the values 1, 2, 3, ... where p = 0.02. X ~ G(0.02)

Find P ( x = 7). Answer: P ( x = 7) = (1 - 0.02) 7-1 × 0.02 = 0.0177.

The probability that the seventh component is the first defect is 0.0177.

The graph of X ~ G(0.02) is:

This graph shows a geometric probability distribution. It consists of bars that peak at the left and slope downwards with each successive bar to the right. The values on the x-axis count the number of computer components tested until the defect is found. The y-axis is scaled from 0 to 0.02 in increments of 0.005.

The y -axis contains the probability of x , where X = the number of computer components tested. Notice that the probabilities decline by a common increment. This increment is the same ratio between each number and is called a geometric progression and thus the name for this probability density function.

The number of components that you would expect to test until you find the first defective one is the mean, μ  = 50 .

The formula for the mean for the random variable defined as number of failures until first success is μ = 1 p = 1 0.02 = 50

See [link] for an example where the geometric random variable is defined as number of trials until first success. The expected value of this formula for the geometric will be different from this version of the distribution.

The formula for the variance is σ 2 = ( 1 p ) ( 1 p − 1 ) = ( 1 0.02 ) ( 1 0.02 − 1 ) = 2,450

The standard deviation is σ = ( 1 p ) ( 1 p − 1 ) = ( 1 0. 02 ) ( 1 0. 02 − 1 ) = 49.5

The lifetime risk of developing pancreatic cancer is about one in 78 (1.28%). Let X = the number of people you ask before one says he or she has pancreatic cancer. The random variable X in this case includes only the number of trials that were failures and does not count the trial that was a success in finding a person who had the disease. The appropriate formula for this random variable is the second one presented above. Then X is a discrete random variable with a geometric distribution: X ~ G ( 1 78 ) or X ~ G (0.0128).

  1. What is the probability of that you ask 9 people before one says he or she has pancreatic cancer? This is asking, what is the probability that you ask 9 people unsuccessfully and the tenth person is a success?
  2. What is the probability that you must ask 20 people?
  3. Find the (i) mean and (ii) standard deviation of X .
  1. P ( x = 9) = (1 - 0.0128) 9 * 0.0128 = 0.0114
  2. P ( x = 20) = (1 - 0.0128) 19 * 0.0128 =0.01
    1. Mean = μ = ( 1 − p ) p = ( 1 − 0.0128 ) 0.0128 = 77.12
    2. Standard Deviation = σ  =  1 − p p 2 = 1 − 0.0128 0.0128 2 ≈ 77.62

Try it

The literacy rate for a nation measures the proportion of people age 15 and over who can read and write. The literacy rate for women in The United Colonies of Independence is 12%. Let X = the number of women you ask until one says that she is literate.

  1. What is the probability distribution of X ?
  2. What is the probability that you ask five women before one says she is literate?
  3. What is the probability that you must ask ten women?
  1. X ~ G (0.12)
  2. P ( x = 5) = 0.0720
  3. P ( x = 10) = 0.0380

A baseball player has a batting average of 0.320. This is the general probability that he gets a hit each time he is at bat.

What is the probability that he gets his first hit in the third trip to bat?

P ( x =3) = (1-0.32) 3-1 × .32 = 0.1480

In this case the sequence is failure, failure success.

How many trips to bat do you expect the hitter to need before getting a hit?

μ = 1 p = 1 0.320 = 3.125 ≈ 3

This is simply the expected value of successes and therefore the mean of the distribution.
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Source:  OpenStax, Introductory statistics. OpenStax CNX. Aug 09, 2016 Download for free at http://legacy.cnx.org/content/col11776/1.26
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