Question

Recall that X ∼ Exp(λ) if the probability density function of X is fX(x) = λe−λx for x ≥ 0. Let X1, . . . , Xn ∼ Exp(λ), where λ is an unknown parameter. Exponential random variables are often used to model the time between rare events, in which case λ is interpreted as the average number of events occurring per unit of time.

Recall that X ~ Exp(A) if the probability density function of X is fx(x)-Ae-Az for z 0. Let Xi, ,Xn ~ Exp(A), where λ is an unknown parameter. Exponential random variables are often used to model the time between rare events, in which case λ is interpreted as the average number of events occurring per unit of time a) Let XI, … , Xn be n observations of an exponential random variable with parameter λ. Prove that the maximum likelihood estimator for A is given by (Xii ^). b) Suppose you observe the time between seven consecutive car accidents to be 25 minutes, 10 minutes, 35 minutes, 80 minutes, 10 minutes, and 20 minutes, respectively. Assume that the time between accidents follows an Exponential distribution. What is the maximum likelihood estimate for the average number of car accidents per day? c) Suppose you are a tow truck driver in San Diego. It is midnight, and you've just finished towing an accident. You want to take your break, but you're concerned about having enough time to eat before the next accident. Someone tells you that, on average, there are 48 accidents per day; they recommend that you model the time until the next accident as being a random draw from Exp(48). What is the problem with this model? Hint: Do you think it will overestimate or underestimate the expected time until the next accident, given that it is midnight?

Answer 1

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