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Let X have mean μx and variance σ, Let Y have mean μy and variance σ.Let Z = X with probability p and Z- Y with probability 1 -p. What are E[Z] and Var[Z]?

This is a homework problem of mine and I don't know how to solve it. Any help would be appreciated.

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The best thing to do here would be to use the tower property for mean and variance.

I.e, call $B$ the random variable which takes value $1$ with probability $p$, and value $0$ with probability $1-p$.

Now, $\mathbb{E}[Z|B = 1] = \mu x$

$\mathbb{E}[Z|B = 0] = \mu y$.

To find $\mathbb{E}[Z]$ we do $\mathbb{E}[Z] = \mathbb{E}[\mathbb{E}[Z|B]] $.

A similar formula exists for the variance: $Var (Z)=\mathbb{E} [Var (Z|B)]+Var (\mathbb{E} [Z| B]))$

Let me know if you need any more help.

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