Prove properties of Expectation for multiple random variables.

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You should first prove expectation is linear (proof is in the discrete case, but in the general case, its linearity of integrals, since expectation is just an integral).

Then, by linearity, prove $var(X) = E[(X - E[X])^2] = E[X^2] - (E[X])^2$ (there is some algebra in the middle). Then, apply linearity and the definition of $Y$ to this to get the result.

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Updated on March 11, 2020

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  • userunknown
    userunknown over 3 years

    Let $X$ and $Y$ be random variables where $Y = cX$ for some constant $c > 0$. Prove that $$E(Y) = cE(x)$$ and $$V(Y) = c^2 V(x)$$ where $E(Y)$ is the expectation of $Y$, $E(x)$ is the expectation of $X$, and $V(Y)$ and $V(x)$ are the variance of $Y$ and $X$ respectively.

    I have a bit of a concept as of why this works. What I was thinking is that since we know that the $E(Y)$ is equal to the sum of $f(k)P(x=k)$ where $f(k)$ is some function (here $Y = cX$)

    From here though I do not know where to go to prove this. Any help is appreciated. Thank you.

  • userunknown
    userunknown over 9 years
    In the proof of expectation being linear... How then do I get from that that E(y) = cE(x)? I see how the proof works but if I am moving the bE(y) over (from the proof) I would have -bE(y) = aE(x) @Batman
  • Batman
    Batman over 9 years
    When you prove expectation is linear, you have shown for any two random variables $X$ and $Y$ and constants $a$ and $b$ that $E[aX+bY] = a E[X] + b E[Y]$. You can just take $b=0$ to get the fact that $E[aX] = a E[X]$.