Find the CDF of Y in terms of CDF of X.

Question

Let $Y$ be a random variable formed from $X$ as $Y=X\mathsf 1_{\{X>0\}} + 2X^2\mathsf 1_{\{X\leqslant 0\}}$.

(a) Compute the cdf of $Y$, $F_Y(y)$, in terms of the cdf of $X$, $F_X(x)$.

(b) Find the pdf of $Y$, $f_Y(y)$, when $f_X(x) = \frac1{\sqrt{2\pi}} e^{-x^2/2}$.

I did the first part in the following way:

For $X>0$:
$$F_Y(y) = \mathbb P(Y\leqslant y) = \mathbb P(X\leqslant Y) = F_X(y).$$

For $X\leqslant 0$: $$ F_Y(y) = \mathbb P(Y\leqslant y) = \mathbb P(2X^2\leqslant y) = F_X\left(\sqrt{y/2})\right). $$

Therefore,

$$ F_Y(y) = F_X(y)\mathsf 1_{(0,\infty)}(x) + F_X\left(\sqrt{y/2}\right)\mathsf 1_{(-\infty)}(y). $$

For the second part I wasn't able to solve as I don't know whether the first part is correct or wrong.

Answer 1

I don't know whether the solution is right or wrong

Fairly close. You have noticed that for positive $y$, the event of $\{Y\leqslant y\}$ is the event of $\{-\sqrt{y/2~}\leqslant X\leqslant 0\}$ or $\{0< X \leqslant y\}$, so $$F_Y(y) = (F_X(y)-F_X(-\surd(y/2)))~\mathbf 1_{y\geqslant 0}$$

[Note: be careful about case sensitivity. The value $y$ and random variable $Y$ should not be confused.]

Now for the second, evaluate the derivative and substitute:$$f_Y(y)=\dfrac{\mathrm d ~~}{\mathrm d y}F_Y(y)$$

Answer 2

Your answer is wrong. The event $(Y\le y)$ is the union of two disjoint events $(0\le X\le y)$ and $(-\sqrt{\frac{y}{2}}\le X\le 0)$. Therefore $F_Y(y)=(F_X(y)-F_X(0))+(F_X(0)-F_X(-\sqrt{\frac{y}{2}})=F_X(y)-F_X(-\sqrt{\frac{y}{2}})$

For the second part $f_Y(y)=\frac{1}{\sqrt{2\pi}}(e^{-\frac{y^2}{2}}+\frac{1}{2\sqrt{2y}}e^{-\frac{y}{4}})$

Note that $F_Y(y)=0$ and $f_Y(y)=0$ for $Y\lt 0$

Note: I started writing this using your handwritten text. However I see that you had partially corrected it for the typed text.