How do you show ? We know that . In other words, controls all the moments.
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Try integration by parts, noting that where F is the CDF. Also, I think you are missing an assumption on the support of .
Originally Posted by Sampras How do you show ? We know that . In other words, controls all the moments. That statement isn't true unless is a nonnegative random variable. Did you really mean to prove the following:
is non-negative. So we know that . Let and . Then and . So So we have: and so the result follows. Is this correct?
Or more generally, one can show that using induction on ? One could use Markov's inequality: ?
Originally Posted by Sampras is non-negative. So we know that . Let and . Then and . So Unfortunately integrating by parts won't help much here because were evaluating a definite integral. Your first line would actually be this: which is obviously no good. I don't know the answer off the top of my head. I'll post back if I have any useful hints.
Originally Posted by Sampras How do you show ? We know that . In other words, controls all the moments. Hint: Now interchange the order of integration in the double integral.
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