The Exponential Distribution (Optional) Quiz

PDF page 348: 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 348: 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 348: 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 348: m = . Therefore, m = = 0.25.

PDF page 348: The standard deviation, σ, is the same as the mean. μ = σ

PDF page 348: The distribution notation is X ~ Exp(m). Therefore, X ~ Exp(0.25).

PDF page 348: The probability density function is f(x) = me . The number e = 2.71828182846... It is a number that is used

PDF page 349: Notice the graph is a declining curve. When x = 0,

PDF page 349: = (0.25)(1) = 0.25 = m. The maximum value on the y-axis is m.

PDF page 350: = P(x < 5) – P(x < 4) = 0.7135 − 0.6321 = 0.0814.

PDF page 350: P(x < k) = 0.50, k = 2.8 minutes (calculator or computer)

PDF page 350: Half of all customers are finished within 2.8 minutes. You can also do the calculation as follows:

PDF page 351: c. From Part b, the median or 50 percentile is 2.8 minutes. The theoretical mean is four minutes. The mean is