Suppose that the random variable X has PDF...

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I've tried to do this question but I'm not sure if I'm correct or not. I also have no idea how to do (c) [not even sure what shortcut he's talking about bit I can ask a classmate]

Here is my file with the question and my attempt at (b) :

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1 Answer

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We do b), first calculating the cdf $F_Y(y)$ of $Y$. Note that $F_Y(y)=0$ if $y\le 0$, and $F_Y(y)=1$ if $y\ge 1$.

Now we calculate the cdf for $0\lt y\lt 1$. We have $$F_Y(y)=\Pr(X^2\le y)=\Pr(-\sqrt{y}\le X\le\sqrt{y}).$$ So $$F_Y(y)=\int_{-\sqrt{y}}^{\sqrt{y}}\frac{x+1}{2}\,dx.$$ Integrate. Conveniently, the $\frac{x}{2}$ part integrates to $0$, and we get $F_Y(y)=\sqrt{y}$ for $0\lt y\lt 1$.

For the density function, differentiate $F_Y(y)$.

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