1. ## Exponentially distributed

Let W be the exponentially distributed waiting time in minutes for the next metro train to arrive.
If P (W < 2) =1/3, compute P (W > 7 | W 5).

2. Originally Posted by affelix
Let W be the exponentially distributed waiting time in minutes for the next metro train to arrive.
If P (W < 2) =1/3, compute P (W > 7 | W 5).
Solve $\int_0^2 \lambda e^{-\lambda x} \, dx = \frac{1}{3}$ for $\lambda$.

$\Pr(W > 7 | W \geq 5) = \frac{\int_7^{+\infty} \lambda e^{-\lambda x} \, dx }{\int_5^{+\infty} \lambda e^{-\lambda x} \, dx }$.

3. But $P(W>7|W>5)=P(W>2)=1-P(W<2)=2/3$

Who's old now and who is MEMORYLESS?
I still think that the point of this problem was to highlight the memoryless property.
That's why I would have assigned it.
It shows that finding lambda is unnecessary.

4. Originally Posted by matheagle
But $P(W>7|W>5)=P(W>2)=1-P(W<2)=2/3$

Who's old now and who is MEMORYLESS?

But .... are these rhetorical question ....? I'm tempted to say that the answer to both is you - the latter because you've clearly forgotten what the reaction of a typical student would be to your reply (assuming a student has been exposed to that property of the exponential distribution) .....

(But you're right, I should have included it as a throw away footnote ....)