Fall 2011 Strategic Practice 6: Section 1 (Exponential Distribution and Memorylessness) - Question 3
Let $X_{1}, ..., X_{n}$ be independent, with $X_{j}\sim Expo(\lambda _{j})$. (They are i.i.d. if all the $\lambda _{j}$'s are equal, but we are not assuming that.) Let M = $min(X_{1}+...+X _{n})$. Show that M ~ $Expo(\lambda _{1}+...+\lambda _{n})$, and interpret this intuitively.
Solution: We can find the distribution of M by considering its survival function P(M > t), since the survival function is 1 minus the CDF. The minimum of the Xj's is greater than t is the same as saying that all of the Xj's are greater than t. Thus, M has the survival function (and the CDF) of an Exponential distribution with parameter  1 + … +  n. Intuitively, it makes sense that M should have a continuous, memoryless distribution (which implies that it's Exponential) and if we interpret lambda{j}'s as rates, it makes sense that M has a combined rate of the n lambda's since we can imagine, for example, X1 as the waiting time for a green car to pass by, X2 as the waiting time for a blue car to pass by, etc., assigning a color to each Xj; then M is the waiting time for a car with any of these colors to pass by.
"Mathematics is the logic of certainty, but statistics is the logic of uncertainty."