How can I get expectations and variances?....
Compute the variance of the exponential density function .
The variance requires us to compute
Using integration by parts, with
This second integral can be done with integration by
parts again, or we can use the fact that this is almost the integral for
the expectation. Namely, we know
and so by dividing through by , we have
Putting this together, we have
Finally, then, the variance is
Find the expectation of , where is uniformly distributed on the interval .
Recall that the PDF associated with is given by for . Thus, the mean is given by
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