Exponentially distributed probability question?

Question

Minor earthquakes frequently occur in the Laurentians. Assume that the "waiting times" in days between earthquakes is exponentially distributed with PDF f(t) = 0.125e^-0.125t. What is the probability that the waiting time for the next earthquake will be at least 5 days?

Answer 1

In general, the PDF of an exponentially distributed random variable T is given by f(t) = a e^(-at), where a>0 is a fixed parameter. So, the probability that T >= t is given by

Integral (t to oo) f(s) ds = Integral (o to oo) a e^(-as) ds = [-e^(-as)] (o to oo) = - (-e^(-at)) = e^(-at), because, since a >0, e^(-as) -> o as s -> oo.

So, F(t) = e^(-at). In your case, a = 0.125. So, for t =5, we get

Prob(T >= 5) = F(5) = e^(0.125*5) =~ 0.535261429

Answer 2

Well it should be straight forward this, we just substitute in 5 for 't'.

f(5) = 0.125 e^(-0.125 * 5)
f(5) = 0.06691