Chapter 5: Continuous Random Variables (41/58) -- Introductory Statistics

Chapter 5: Continuous Random Variables 5.3 The Exponential Distribution Learning Objectives By the end of this section, the student should be able to: - calculate exponential distribution and the probability density function. The exponential distribution is often concerned with the amount of time until some specific event occurs. For example, the amount of time (beginning now) until an earthquake occurs has an exponential distribution. Other examples include the length, in minutes, of long distance business telephone calls, and the amount of time, in months, a car battery lasts. It can be shown, too, that the value of the change that you have in your pocket or purse approximately follows an exponential distribution. Values for an exponential random variable occur in the following way. There are fewer large values and more small values. For example, the amount of money customers spend in one trip to the supermarket follows an exponential distribution. The exponential distribution is widely used in the field of reliability. Reliability deals with the amount of time a product lasts. Example Let X = amount of time (in minutes) a postal clerk spends with his or her customer. The time is known to have an exponential distribution with the average amount of time equal to four minutes. X is a continuous random variable since time is measured. It is given that [latex]\mu = 4[/latex] minutes. To do any calculations, you must know m, the decay parameter. [latex]m=\frac{1}{\mu }[/latex]. Therefore, [latex]m=\frac{1}{4}=0.25.[/latex] The standard deviation, [latex]\sigma[/latex], is the same as the mean. [latex]\mu = \sigma[/latex] The distribution notation is [latex]X \sim Exp(m)[/latex]. Therefore, [latex]X \sim Exp(0.25)[/latex]. The probability density function is [latex]f(x) = me-mx[/latex]. The number [latex]e = 2.71828182846...[/latex] It is a number that is used often in mathematics. Scientific calculators have the key “[latex]e^{x}[/latex].” If you enter one for x, the calculator will display the value e. The curve is: [latex]f(x) = 0.25e–0.25x[/latex] where [latex]x[/latex] is at least zero and [latex]m = 0.25[/latex]. For example, [latex]f(5)=0.25e - (0.25)(5) = 0.072[/latex]. The postal clerk spends five minutes with the customers. The graph is as follows: Notice the graph is a declining curve. When [latex]x = 0[/latex], [latex]f(x) = 0.25e(−0.25)(0) = (0.25)(1) = 0.25 = m[/latex]. The maximum value on the y-axis is m. Your Turn! The amount of time spouses shop for anniversary cards can be modeled by an exponential distribution with the average amount of time equal to eight minutes. Write the distribution, state the probability density function, and graph the distribution. Solution [latex]X \sim Exp(0.125)[/latex]; [latex]f(x)=0.125 e ^{-0.125x}[/latex]; Example a. Using the information in the graph from the first example (shown below), find the probability that a clerk spends four to five minutes with a randomly selected customer. Solution a. Fin