poisson distribution with a mean that is a uniform random variable

paulrb

It's been a while since I've taken probability so I'm not sure if I have this right:

Question: "Suppose y|x is poisson with mean x and X ~ Uniform[0,5]. compute Pr[Y>=1]"

Here is what I did: E[Y] = E[E(Y|X)] = E[X] = 2.5 = the parameter for poisson

Pr[Y>=1] = 1 - Pr[Y = 0]

= 1 - e^(-2.5)

Is this right? It seems overly simple so I'm not sure...