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I have a question regarding finding the following joint distribution.

Let $p \sim U[0,1]$, standard uniform distribution.

The random variable $X$ is defined as $X = 2$ with probability $p$ and $X = 0$ with probability $(1-p)$.

The random variable $Y$ is defined as $Y = 4p$.

The question is to find the $cov(X,Y)$ and the joint distribution of $X$ and $Y$

This is what I have done so far:

Since $p \sim U[0,1]$, then $Y \sim U[0,4]$ since $Y = 4p$. Then I used the formula $$Cov(X,Y) = E(XY) - E(X)E(Y) $$ In this case, $E(X)=2p$ and $E(Y)=2$. However, I get stuck when attempting to find $E(XY)$ along with the the joint distribution. Any suggestion or help would be extremely appreciated!!

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When approaching this kind of questions your main tools are Bayes rule and the law of total expectation and total probability. Notice that the data about X in the question is infact about the conditioned RV X|p For the covariance apply the law of total expectation: $E[XY]=E[E[XY|p]]=E[E[X4p|p]]=E[4pE[X|p]]=E[8p^2]=8(0.25+\frac{1}{12})=\frac{8}{3}$ and notice that $E[X]=E[E[X|p]]=E[2p]=1$ For the PDF, first apply the Bayes rule: $f_{X,Y}(a,b)=P(X=a|Y=b)f_{Y}(b)$ The PDF of $Y$ is known as you mentioned that it follows the distribution $U(0,4)$. Then, $P(X=0|Y=b)=1-0.25b,P(X=2|Y=b)=0.25b$.

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