Say that X has an exponential distribution with unknown lambda. Then I'm told that lambda has a gamma distribution. Now I'm asked to find the density of X. How do I approach this problem?
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It's called a mixture of distributions, and you solve it using integrals so we know and (assuming the parameters are alpha and beta).
How would I find E(X) and Var(X) without using the pmf just calculated?
I don't think you can. Since E(x) = and same idea for Var(X). Also, since f(x) is continous, it's a PDF, not a PMF.
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