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).
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
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