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Exponential Probability Distribution for Beginning Statistics

By — McGraw-Hill Professional
Updated on Aug 12, 2011

Practice problems for these concepts can be found at:

Continuous Random Variables and Their Probability Distributions Solved Problems for Beginning Statistics

The exponential probability distribution is a continuous probability distribution that is useful in describing the time it takes to complete some task. The pdf for an exponential probability distribution is given by formula (6.7), where μ is the mean of the probability distribution and e = 2.71828 to five decimal places.

The graph for the pdf of a typical exponential distribution is shown in Fig. 6-30.

Exponential Probability Distribution

EXAMPLE 6.15 The exponential random variable can be used to describe the following characteristics: the time between logins on the internet, the time between arrests for convicted felons, the lifetimes of electronic devices, and the shelf life of fat free chips.

Probabilities For The Exponential Probability Distribution

A standardized table of probabilities does not exist for the exponential distribution and to find areas under the exponential distribution curve requires the use of calculus. However, formula (6.8) is useful in solving many problems involving the exponential distribution.

Probabilities For The Exponential Probability Distribution

EXAMPLE 6.16 Suppose the time till death after infection with HIV, the AIDS virus, is exponentially distributed with mean equal to 8 years. If X represents the time till death after infection with HIV, then the percent who die within five years after infection with HIV is found by multiplying P(X ≤ 5) by 100. The probability is found as follows: P(X ≤ 5) = 1 – e–.625 = 1 – .535 = .465. Using the MINITAB pull-down menu Calc ⇒ Probability distributions ⇒ Exponential, the exponential distribution dialog box is given. In the dialog box check cumulative distribution, mean = 8 and input constant equal to 5. The following output results.

Cumulative Distribution Function

Exponential with mean = 8

x     P(X <= x)

5     0.464739

To find the percent who live more than 10 years, we multiply P(X > 10) by 100. In order to utilize formula (6.8), we use the complementary rule for probabilities. This rule allows us to write P(X > 10) as follows:

P(X > 10) = 1 – P(X ≤ 10) = 1 – (1 – e–1.25) = e–1.25 = .287

That is, 28.7% of the individuals live more than 10 years after infection. This probability is shown as the shaded area in Fig. 6-32.

Probabilities For The Exponential Probability Distribution

To find the percent who live between 2 and 4 years after infection, we multiply P(2 < X < 4) by 100. To use formula (6.8) to find this probability, we express P(2 < X < 4) as follows:

P(2 < X < 4) = P(X < 4) – P(X < 2)= (1 – e–.5) – (1 – e–.25) = e–.25 – e–.5 = .172

That is, 17.2% live between 2 and 4 years after infection. This probability is shown as the shaded area in Fig. 6-33.

Probabilities For The Exponential Probability Distribution

Practice problems for these concepts can be found at:

Continuous Random Variables and Their Probability Distributions Solved Problems for Beginning Statistics

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