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Dear friends I am Bernardo I am trying to derive the variance of an estimator, but I need help in some concepts. I will relate to a very simple way: suppose that we have three random variables: $X, Y, Z$ 2 cases: case 1: $X,Y$ dependents and $Z$ independent of both. case 2: all dependents

I want the variance of $X+Y\cdot Z$

I will greatly appreciate your help. Thanks in advance Bernardo

Giovanni
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Bernardo
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1 Answers1

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Hint for case $1)$ : Recall that $\mathrm{V}[X]$ can be rewritten as

$\mathrm{V}[X]=\mathrm{E}[X^2]+(\mathrm{E}[X])^2$

And that if two random variables $X$ and $Y$are independent, then:

$\mathrm{E}[XY]=\mathrm{E}[X]\mathrm{E}[Y]$

Using the linearity of the expectation operator, that is:

$\mathrm{E}[\alpha X+\beta Y]=\alpha \mathrm{E}[X]+\beta \mathrm{E}[Y]$

you should be able to continue from that.