How can I calculate E[∑Xi=1I{Yi≤Yn+1}]\mathrm{E}\!\left[\sum _{i=1}^X I_{\{Y_i\leq Y_{n+1}\}}\right]?

Suppose that Y1,,Yn+1 is a random sample from a continuous distribution function F. LetXUniform{1,,n} be independent of the Yi‘s. How can I compute E[Xi=1I{YiYn+1}]?


Here is an alternative answer to @Lucas’ using the law of iterated expectations:


The third step follows from independence of Yi and Yn+1 from X; the fourth step is again an application of the law of iterated expectations; the last step is simply an application of the formula for the expectation of a discrete uniform random variable.

By inverting the order of integration, we derive the remaining expectation:


which implies E[F(Yn+1)]=12. Hence:


Source : Link , Question Author : hadi , Answer Author : Daneel Olivaw

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