Question

Suppose that X|λ is an exponential random variable with parameter λ and that λ|p is geometric with parameter p. Further suppose that p is uniform between zero and one. Determine the pdf for the random variable X and compute E(X).

Answer #1

Therefore ..............................................(since ).....................(1)

Therefore ..................................................(Since )......................(2)

Therefore ........................................................(since ).........(3)

Using the law of total expectation we have

Using this in (2) we use it to calculate

.......................................from (3)

........................................................Since expectation of a constant is constant itself

Using the law again in (1)

..................................Since expectation of a constant is constant itself

*Please let me know if you have any doubts by mentioning them
in the comments section. Also if this helped please give a thumps
up.*

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