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My problem is something like this problem We have $ X = Pois (\lambda)$ and $Y|X = binom (X , p)$ Where p is a known constant. And I want to calculate the correlation of $X$ and $Y$. I read the answer to that question but I really have no idea what to do for this question. I actually can't calculate covariance even after reading that. Because I don't know what is actually the distribution of $Y$ in order to calculate $COV(X,Y)=\sum P(X,Y) * (X-\mu_x)(Y-\mu_Y)$. Also I can't Calculate Standard deviation of $Y$ because I don't Know the distribution. So I can't calculate $COR(X,Y) = COV(X,Y)/\sigma_x\sigma_y$. What should I do?

I am very new to statistics and probability and I really don't have any idea where should I start when solving these problems. Note that I am not allowed to use iterated expectation in this question.

amir na
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    This is truly a minor variation of the referenced question: just follow exactly the steps outlined there. – whuber Dec 12 '19 at 20:33
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    @whuber . I am not allowed to use law of iterated expectations for this question. the question is from an exam and the law of iterated expectations is not taught at time of the exam; so it could be solved without it. For the correlation part, I don't know how to get $\sigma_y$; Because I don't know its distribution. – amir na Dec 12 '19 at 20:39
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    Then it's crucial that you explicitly give those restrictions in your question, as well as applying the [tag:self-study] tag, for otherwise the community will close it as a duplicate. – whuber Dec 12 '19 at 20:40
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    I found an answer to your question about finding the distribution of $Y:$ it's at https://stats.stackexchange.com/questions/288807. ("What is the distribution of the number of successes in a Poisson process followed by a series of bernoulli trials, where the number of trials is determined by the result of the poisson process?") – whuber Dec 12 '19 at 20:42

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