The Exponential Distribution Quiz

PDF page 257: 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 of time, 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, marketing studies have shown that 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. The random variable for the exponential distribution is continuous and often measures a passage of time, although it can be used in other applications. Typical questions may be, “what is the probability that some event will occur within the next x hours or days, or what is the probability that some event will occur between x hours and x hours, or what is the 1 2 probability that the event will take more than x hours to perform?” In short, the random variable X equals (a) the time 1 between events or (b) the passage of time to complete an action, e.g. wait on a customer. The probability density function is given by: 1 − x 1 μ f (x) = e μ where μ is the historical average waiting time. and has a mean and standard deviation of 1/μ.

PDF page 257: 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 of time, 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, marketing studies have shown that 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. The random variable for the exponential distribution is continuous and often measures a passage of time, although it can be used in other applications. Typical questions may be, “what is the probability that some event will occur within the next x hours or days, or what is the probability that some event will occur between x hours and x hours, or what is the 1 2 probability that the event will take more than x hours to perform?” In short, the random variable X equals (a) the time 1 between events or (b) the passage of time to complete an action, e.g. wait on a customer. The probability density function is given by: 1 − x 1 μ f (x) = e μ where μ is the historical average waiting time. and has a mean and standard deviation of 1/μ.

PDF page 266: of an exponential random variable. It is also 1

PDF page 267: 1 If X has an exponential distribution with mean μ, then the decay parameter is m = . The probability density function μ

PDF page 258: m = . Therefore, m = = 0.25 .

PDF page 258: f(x) = 0.25e where x is at least zero and m = 0.25.

PDF page 258: = (0.25)(1) = 0.25 = m. The maximum value on the y-axis is always m, one divided by the

PDF page 260: P(x > 7) = e = 0.4966. The probability that a computer part lasts more than seven years is 0.4966.

PDF page 260: after the other would last, on the average, (5)(10) = 50 years.

PDF page 261: ) = 0.6671 – 0.5934 = 0.0737. The probability

PDF page 261: that a computer part lasts between nine and 11 years is 0.0737.

PDF page 262: Recall that the amount of time between customers for the postal clerk discussed earlier is exponentially distributed with a mean of two minutes. Suppose that five minutes have elapsed since the last customer arrived. Since an unusually long amount of time has now elapsed, it would seem to be more likely for a customer to arrive within the next minute. With the exponential distribution, this is not the case–the additional time spent waiting for the next customer does not depend on how much time has already elapsed since the last customer. This is referred to as the memoryless property. The exponential and geometric probability density functions are the only probability functions that have the memoryless property. Specifically,

PDF page 263: P (X > r + t | X > r) = P (X > t) for all r ≥ 0 and t ≥ 0

PDF page 266: μ . memoryless property For an exponential random variable X, the memoryless property is the statement that knowledge of what has occurred in the past has no effect on future probabilities. This means that the probability that X exceeds x + t, given that it has exceeded x, is the same as the probability that X would exceed t if we had no knowledge about it. In symbols we say that P(X > x + t|X > x) = P(X > t). Poisson distribution If there is a known average of μ events occurring per unit time, and these events are independent of each other, then the number of events X occurring in one unit of time has the Poisson distribution. The probability x −μ

PDF page 261: a. Since we expect 30 customers to arrive per hour (60 minutes), we expect on average one customer to arrive

PDF page 261: every two minutes on average.

PDF page 262: c. Let X = the time between arrivals, in minutes. By part a, μ = 2, so m = = 0.5.

PDF page 262: Therefore P(X < 1) = 1 – e = 0.3935.