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Let $X$, $Y$ and $Z$ be independent random variables. Let $X$ be Bernoulli distributed on $\{0,1\}$ with success parameter $p_0$ and let $Y$ be Poisson distributed with parameter $\lambda$ and let $Z$ be Poisson distributed with parameter $\mu$.

(a) Calculate the distribution, the expected value and the variance of $XY$.

(b) Determine the Covariance and the correlation between $XY$ and $XZ$.

For question (a) :

We have that $$P(X=0)=1-p_0 \ \text{ and} \ P(X=1)=p_0$$ and $$P(Y=k)=\frac{\lambda^k}{k!}\cdot e^{-\lambda}$$

Sodo we get that $$P(XY=k)=P(XY=k|X=0)P(X=0)+P(XY=k|X=1)P(X=1)$$ If $k=0$ then \begin{align*}P(XY=0)&=P(XY=0|X=0)P(X=0)+P(XY=0|X=1)P(X=1)\\ & =1-p_0+e^{-\lambda}\end{align*} If $k\neq 0$ then \begin{align*}P(XY=k)&=P(XY=k|X=0)P(X=0)+P(XY=k|X=1)P(X=1)\\ & =0+\frac{\lambda^k}{k!}\cdot e^{-\lambda}\cdot p_0\\ & = \frac{\lambda^k}{k!}\cdot e^{-\lambda}\cdot p_0\end{align*}

Is that correct? :unsure: