Poisson Process Question

Suppose X(t) is a Poisson process with rate 2. Let W[sub:3k16n7lq]n[/sub:3k16n7lq] be the waiting time for the nth event, n=1,2,... Find E(X(2)|W[sub:3k16n7lq]2[/sub:3k16n7lq]>1).

1. Is it true to say that {W[sub:3k16n7lq]2[/sub:3k16n7lq]>1} = {X(1)<2}, so E(X(2)|W[sub:3k16n7lq]2[/sub:3k16n7lq]>1) = E(X(2)|X(1)<2)?
2. Are my subsequent steps correct?
E(X(2)|X(1)<2) = {P(X(1)=0)|X(1)<2) * E[X(2)|X(1)=0]} + {P(X(1)=1)|X(1)<2) * E[X(2)|X(1)=1]}
Since
P(X(1)=0)|X(1)<2) = 1/3
E[X(2)|X(1)=0] = 0 + (2-1)(2) = 2
P(X(1)=1)|X(1)<2) = 2/3
E[X(2)|X(1)=0] = 1 + (2-1)(2) = 3

The required answer is 8/3 ?

1. Yes
2. Not too sure about the correctness of the first step, but I think it's clearer to rewrite the conditional expectation as

\(\displaystyle E[X(2)|X(1)<2] = E[(X(2)-X(1))+X(1)|X(1)<2]\)

Then by linearity of expectation and and independence of increments of a Poisson process, we have the equation to be equal to

\(\displaystyle E[X(2)-X(1)] + E[X(1)|X(1)<2]\)

The rest should be straightforward and I believe the answer is 8/3.