Assume that events occurring in time follow a Poisson distribution with rate . Let T be the

Assume that events occurring in time follow a Poisson distribution with rate λ. Let T be the amount of time, in seconds, that elapses between two events. This exercise will show how to compute probabilities involving T.

a. Let X be the number of events that occur in a 1-second interval. Show that P(X = 0) = e−λ.

b. Explain why X = 0 is the same as T > 1.

c. Show that P(T > 1) = e−λ.

d. Let X be the number of events that occur in a 2-second interval. Show that P(X = 0) = e−2λ.

e. Explain why X = 0 is the same as T > 2.

f. Now let t be any amount of time, and let X be the number of events that occur in an interval of length t seconds. Show that P(X = 0) = e−λt .

g. Explain why X = 0 is the same as T > t.

h. Show that P(T > t) = e−λt.