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Consider a random variable $X$ which can take only non zero integer values from -20 to +20, and whose probability distribution is symmetric around 0. Suppose the function $f(x)$ is the probability mass function of X. Now consider the random variable $Y= X^2$. Derive the relationship between g(y) and f(y).

Since $Y= X^2$, do we have g(y) = f($\sqrt y$)? Can we derive probabilities like this?

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  • $\begingroup$ Yes, if we do it correctly. If $y>0$ then $g(y)=2f(\sqrt y)$ $\endgroup$ – drhab Jun 18 '15 at 16:00
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Note that $Y$ takes values in $\{k^2\mid k\in\{-20,\dots,-1,0,1,\dots,20\}\}=\{k^2\mid k\in\{0,1,\dots,20\}\}$ and:

$$g(0)=P(Y=0)=P(X^2=0)=P(X=0)=f(0)$$

For $x\in\{1,2,\dots,20\}$:$$g(x^2)=P(Y=x^2)=P(X^2=x^2)=P(X=x\wedge X=-x)=$$$$P(X=x)+P(X=-x)=f(x)+f(-x)=2f(x)$$

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