Here's a problem I am stuck on:
Let $X$ and $Y$ be independent random variables such that $X$ is Bernoulli-distributed with $p=1/2$, and $Y$ is uniformly distributed on the interval $[0,1]$. Then:
- What is the CDF and PDF of $X+Y$ ?
- Does the PDF of $XY$ exist?
- What is the CDF of $XY$?
I tried finding the CDF of $X+Y$ by conditioning on $X$ as per this answer, but could not get any further. Can anyone show me what to do, or how to do this?
On 1)
Let $W:=X+Y$. Then:
$$F_{W}\left(w\right)=P\left(X+Y\leq w\mid X=0\right)P\left(X=0\right)+P\left(X+Y\leq w\mid X=1\right)P\left(X=1\right)=$$$$\frac{1}{2}F_{Y}\left(w\right)+\frac{1}{2}F_{Y}\left(w-1\right)$$
Here $F_{Y}$ is well known to you and knowing CDF $F_{W}$ you can find PDF $f_{W}$.
On 2)
$X=0\Rightarrow XY=0$ so that $P\left\{ XY=0\right\} \geq P\left\{ X=0\right\} \geq\frac{1}{2}$. Draw your conclusions about the existence of a PDF.
On 3)
Let $V:=XY$. Then:
$$F_{V}\left(v\right)=P\left(XY\leq v\mid X=0\right)P\left(X=0\right)+P\left(XY\leq v\mid X=1\right)P\left(X=1\right)=$$$$\frac{1}{2}P\left(0\leq v\right)+\frac{1}{2}F_{Y}\left(v\right)$$
Here $P\left(0\leq v\right)=0$ if $v<0$ and $P\left(0\leq v\right)=1$ otherwise.