The Exponential Distribution (Optional) Quiz

PDF page 326: 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. There are more people who spend small amounts of money and fewer people who spend large amounts of money. Exponential distributions are commonly used in calculations of product reliability, or the length of time a product lasts.

PDF page 326: 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. There are more people who spend small amounts of money and fewer people who spend large amounts of money. Exponential distributions are commonly used in calculations of product reliability, or the length of time a product lasts.

PDF page 326: The standard deviation, σ, is the same as the mean. μ = σ

PDF page 326: 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 μ = 4 minutes. To do any calculations, you must know m, the decay parameter. . Therefore,

PDF page 326: The distribution notation is X ~ Exp(m). Therefore, X ~ Exp(0.25). -mx The probability density function is f(x) = me . The number e = 2.71828182846... It is a number that is used often in x mathematics. Scientific calculators have the key "e ." If you enter one for x, the calculator will display the value e. The curve is –0.25x f(x) = 0.25e where x is at least zero and m = 0.25. (−0.25)(5) For example, f(5) = 0.25e = 0.072. The probability that the postal clerk spends five minutes with the customers is 0.072. The graph is as follows:

PDF page 326: The distribution notation is X ~ Exp(m). Therefore, X ~ Exp(0.25). -mx The probability density function is f(x) = me . The number e = 2.71828182846... It is a number that is used often in x mathematics. Scientific calculators have the key "e ." If you enter one for x, the calculator will display the value e. The curve is –0.25x f(x) = 0.25e where x is at least zero and m = 0.25. (−0.25)(5) For example, f(5) = 0.25e = 0.072. The probability that the postal clerk spends five minutes with the customers is 0.072. The graph is as follows:

PDF page 327: Notice the graph is a declining curve. When x = 0, (−0.25)(0) f(x) = 0.25e

PDF page 327: = (0.25)(1) = 0.25 = m. The maximum value on the y-axis is m.

PDF page 328: The probability that a postal clerk spends four to five minutes with a randomly selected customer is P(4 < x < 5) = P(x < 5)

PDF page 328: – P(x < 4) = 0.7135 − 0.6321 = 0.0814.

PDF page 328: P(x < k) = 0.50, k = 2.8 minutes (calculator or computer)

PDF page 328: Half of all customers are finished within 2.8 minutes. You can also do the calculation as follows:

PDF page 329: c. From Part b, the median or 50 percentile is 2.8 minutes. The theoretical mean is four minutes. The mean is larger.

PDF page 329: (–0.1)(7) P(x > 7) = e = 0.4966. The probability that a computer part lasts more than seven years is 0.4966.