# exponential RV til bus arrives

Suppose that you are waiting at a bus stop. The waiting time until a bus arrives is $$T$$ where $$T$$ is an exponentially distributed random variable with parameter $$λ$$ i.e. $$P(T≤t)=1−e^{−λt}, ∀t≥0$$.
(a) Given that you have already waited $$r$$ seconds, what is the probability that the bus will not arrive within $$d$$ more seconds?
(b) What is the average waiting time for the bus i.e. the expected value of $$T$$? Hint: Recall that one way to solve $$\int u \, dv$$ is by integration by parts.

My attempt:

(a) If we've waited $$r$$ seconds and seeing if the bus will not arrive in $$d$$ more seconds, then we're calculating $$1-P(T≤r+d)$$ which is $$1-(1−e^{−λ(r+d)}) = e^{−λ(r+d)}$$

(b) The PDF of an exponential RV is $$f(x)=λe^{-λx}$$. The average waiting time would be $$\int_0^\infty xλe^{-λx} \, dx$$ which, after integration, equals $$\frac{1}{λ}$$.

Is this correct?

But (a) asks for a conditional probability. Recall that $$\Pr(A\mid B) = \frac{\Pr(A\ \&\ B)}{\Pr(B)}.$$
• @IrCa : Yes. $\qquad$ Commented Apr 8, 2019 at 5:01