Let $d \in \mathbb{R} $ and let $X$ be a random variable with support $(0,\infty$) and $Y$ be a random variable defined as follows
$$
Y =
\begin{cases}
0 & , X \le d \
X & , X d
\end{cases}
$$
Find the cdf and pdf of Y.

Question

Let $d \in \mathbb{R} $ and let $X$ be a random variable with support $(0,\infty$) and $Y$ be a random variable defined as follows
$$
 Y =
  \begin{cases}
   0 & , X \le d \
   X & , X > d
  \end{cases}
$$
Find the cdf and pdf of Y.

kirbykirby · Answer

I have the solution written as 

cdf: $$
 F_Y(y) =
  \begin{cases}
   F_X(d) & , y=0 \
   F_X(d)       &, 0<y<d \
   F_X(y) & , y\ge d \
  \end{cases}
$$
pdf: $$
 f_Y(y) =
  \begin{cases}
   F_X(d) & , y=0 \
   0      &, 0<y<d \
   f_X(y) & , y\ge d \
  \end{cases}
$$

I'm hoping for some clarification on how to find these! This is related to mixed random variables (with continuous and discrete parts) which I'm not too familiar with!

kirbykirby · Answer

I'm guessing it has to do with manipulating the expression $P(Y\le y)$ though I'm not to clear on it in this case

zarkon · Answer

$$F_Y(y)=P(Y\le y)=P(Y\le y,X\le d)+P(Y\le y,X>d)$$

$$P(Y\le y|X\le d)P(X\le d)+P(Y\le y|X>d)P(X>d)$$