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Calculate the mean and variance of the discrete random variable X with:

$f(x)=\frac{1}{2^x}$ if $x=1, 2, 3, \ldots$, $f(x)=0$ otherwise.

Then, $E[X]=1(1/2)+2(1/2^2)+3(1/2^3)+4(1/2^4)+\ldots$

But now, I don't know how to express this in order to continue with my calculations.

Max
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  • So... you want $\sum_{n=1}^{\infty}\frac{n}{2^n}$? If so, just find the general form of $\sum nx^n$ by differentiating $\sum x^n$ and putting $x=\frac{1}{2}$. – TheBestMagician Oct 14 '21 at 00:57
  • Or consider $E[X]-\frac12E[X]$ and then $(E[X]-\frac12E[X]) -\frac12(E[X]-\frac12E[X])$ – Henry Oct 14 '21 at 01:03
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    A google search for 'mean and variance of geometric distribution' would lead you to the countless previous posts of this question on this site. – StubbornAtom Oct 14 '21 at 07:53

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