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The number of defects per yard, denoted by X, for a certain fabric is known to have a Poisson distribution with parameter $\lambda$. However, $\lambda$ is not known, and is itself assumed to be random and exponentially distributed with mean 2. (a) Find the (unconditional) pmf of X. (b) Compute the mean EX and the variance Var(x) without using the pmf computed in part (a).

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Sounds like a nice homework problem copied here with all the imperative directions "Find this" "Compute that" as is. What are your ideas on how to go about solving this problem? Where are you stuck? –  Dilip Sarwate Nov 5 '12 at 13:26
@DilipSarwate I don't know how to begin the problem. Any suggestions would be very helpful. –  woaini Nov 5 '12 at 13:42
Hint: You are told the conditional probability of the event $\{X=k\}$ given the values of $\lambda$; you are asked for the unconditional probability. The standard way of tackling problems like this is to use the law of total probability $$P(A) = \sum_i P(A\mid B_i)P(B_i)$$ where the $B_i$ are a (countable) partition of the sample space, or, when the conditioning is on the value of a continuous random variable $X$, $$P(A) = \int_{-\infty}^\infty P(A\mid X=x)f_X(x)\,\mathrm dx.$$ –  Dilip Sarwate Nov 5 '12 at 18:00
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