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In particular in the following link, Find the PDF of $(y\cos(\theta), y\sin(\theta))$, the expected value was calculated before the pdf was given and I do not see how that is possible. Can someone care to explain?

Jhon Doe
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  • Indirect answer: whenever you are asked to find an expectation then always first try to find it without making use of PDF, PMF, CDF or whatsoever. Very often that is very fruitful. – drhab Apr 17 '20 at 10:47
  • thanks but in the link provided how was the pdf determined form the expected value? – Jhon Doe Apr 17 '20 at 10:54
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    If on forehand it is known what sort of distribution you are dealing with then finding PDF of PMF comes to finding parameters. For instance if $N$ has Poisson distribution then we already have quite some info about the PMF. In that case the info can be made complete by calculation of $\lambda=\mathbb EN$. – drhab Apr 17 '20 at 11:00
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    In the link that you provided indeed the pdfs of the original random variables $\theta, y$ are used: check the formulas that are used to calculate the expectation. The key of that comment, however, is that it is not the expectation of the actual r.v. that is computed but rather the expectation of any bounded function of this random variable. This then gives you the distribution is this case, as equality of expecation of any bounded function $f$ means equality in distribution. (i.e. $E[f(X)] = E[f(Y)], \forall f \implies X \sim Y $ – a_student Apr 17 '20 at 11:05
  • @a_student Hi thanks for your comment. But how do you prove the statement regarding the equality of expectation of any bounded function. If you can provide the proof or link to the proof I can accept it as an answer for you. – Jhon Doe Apr 17 '20 at 11:10
  • If you are confused about the linked answer to your question, then it is based on this principle. – StubbornAtom Apr 17 '20 at 11:11
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    It's just like they told you if $E[f(X)]=E[f(Y)] \forall f$ implies $X$ and $Y$ have the same distribution, take $f=1_B$ you will have $P_X(B)=P_Y(B),$ I also used substitution technique to compute it. – mathex Apr 17 '20 at 11:25

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