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I know that if $X,Y$ are independent, then $E(XY) = E(X)E(Y)$ holds. But the reverse is not true. I'd like to know that are there any examples to show the fact?

pixas
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  • Please edit title to match your actual question. It's hard to say what if anything your title is asking but it can't be what you meant to ask – bof Nov 12 '20 at 06:53

2 Answers2

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Let $X$ and $Z$ be independent random variables such that $X$ takes $0$ and $1$ with probability $1/2$ each and $Z$ takes $-1$ and $1$ with probability $1/2$ each.

Then $XZ$ and $X$ satisfy the condition you have given but are not independent.

Amit Rajaraman
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Let $X$ be a random variable that takes value $0$ with probability $1/2$ and $1$ with probability $1/2$. Let $T$ be a random variable, independent of $X$, that takes the value $−1$ with probability $1/2$, and takes the value $1$ with probability $1/2$. And let $Y = XT$.

We have that $E[X]=1/2$ and $E[Y] =0$. I'll leave it to you to check that $E[XY] = [X^2 T]=0$, which would mean $E[XY]=E[X]E[Y]$.

Now the question is, are $X$ and $Y$ independent? What's the probability that $Y=1?$ What's the probability that $Y=1$ given that $X=0?$ Conditioning on $X$ affects $Y$, so they're not independent.

Math Helper
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