Can you help me find the following:
$$E [W_1 \mid X(3) = 1]$$
$W_1$ is the time that the first event occurs, and $X(3) = 1$ signifies that one event occured before time 3. The rate of this question is lambda.
Here is what I have done so far:
I know that $$P(W_1 \le s \mid X(3) = 1) = P(X(s) \ge 1 \mid X(3) = 1) = 1 - P(X(s) = 0 \mid X(3) = 1) = 1-s/3$$
Differentiating this gives me the density $-1/3$.
Taking the integral from s to 3 of $-1/3$ gives me $1 - s/3$. Is this correct? I believe the answer should not be in terms of s so this leads me to believe I am wrong. Help would be great. Thanks a bunch.