Submitted by Luke_344 on Mon, 05/29/2017 - 20:37

In the STEP III Statistics module, when the distribution of $Y$ is being worked out given $Y=X^2$, I'm having some trouble understanding the following lines:

$$

\begin{aligned}

\mathrm{F}_Y(y)&=\mathrm{P}(Y \leqslant y)\\

&=\mathrm{P}(X^2 \leqslant y)\\

&=\mathrm{P}(X \leqslant \sqrt{y})\\

&=\int_{-\infty}^{\sqrt{y}} \mathrm{f}(t)\mathrm{d}t

\end{aligned}

$$

I'm confused because I don't see how it isn't as follows:

$$

\begin{aligned}

\mathrm{F}_Y(y)&=\mathrm{P}(Y \leqslant y)\\

&=\mathrm{P}(X^2 \leqslant y)\\

&=\mathrm{P}(-\sqrt{y} \leqslant X \leqslant \sqrt{y})\\

&=\int_{-\sqrt{y}}^{\sqrt{y}} \mathrm{f}(t)\mathrm{d}t

\end{aligned}

$$

I'd be grateful for any clarification you could provide about this situation.

Thanks,

Luke.

## Mistake!

Sorry! Will get it fixed.

Thanks for pointing it out.