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**[Solved] Independance of RVs with same distribution**

Hey all,

Let's say we have two Gaussian random variables X, Y, each with zero mean and unit variance. Is it correct to say that [tex]P(X|Y) = P(X)[/tex]?

In other words, suppose that we want to compute the expectation of their product [tex]\operatorname{E}[XY][/tex]. Is the following correct? I.e. does their joint distribution factorise?

[tex]E[XY] = \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}x y\: p(x) p(y)\: dx dy[/tex]

[tex]= \int_{-\infty}^{\infty}x \:p(x)\: \operatorname{d} x \int_{-\infty}^{\infty} y \: p(y) \: \operatorname{d} y[/tex]

[tex]= \operatorname{E}[X]\operatorname{E}[Y] \nonumber[/tex]

Many Thanks.

Update

I have now figured out the answer to the above questions. I'll post it here for anyone who is interested.

If X and Y have the same distribution, then we can write [tex]P(X|X) = 1 \neq P(X)[/tex].

Now looking again at expectations. From the above, we have that

[tex]E[XY]=E[X^2][/tex]

[tex]=\int_{-\infty}^{\infty}x^2 p(x^2) \: dx[/tex]

similarly giving a negative answer for the expectation of the product.

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