Product of a exponential and discrete Distribution Let $U \sim e(1)$ and $V$ a discrete random variable independent of $U$ such that
$p_V(v)=1/2$ if $v \in \{-1,1\}$ and $p_V(v)=0$ otherwise.
Problem:
Let $W=UV$. Find the distribution function of $W$ $\forall w\in \mathbb{R}$.
My try:
\begin{align}
P(W\leq w)&=P(UV\leq w)=P(U\leq w \mid V=1)P(V=1)+P(-U\leq w \mid V=-1)P(V=-1)\\
&= P(U\leq w)\frac{1}{2}+P(U\geq -w)\frac{1}{2}\\
&=P(U\leq w)\frac{1}{2}+(1-P(U\leq-w)\frac{1}{2}\\
&=\left(\int_0^we^{-u} \, du\right)\frac{1}{2}+\left(1-\int_0^{-w} e^{-u} \, du \right) \frac{1}{2}\\
&=(1-e^{-w})\frac{1}{2}+\frac{1}{2}e^w = F_W(w)
\end{align}
The correct answer in my book is the following
$$F_W(w)=\frac{1}{2}(1-e^{-w})+\frac{1}{2}, w>0$$ and
$$F_W(w)=\frac{1}{2}e^{w}, w\leq 0$$
Can anyone tell me what i am doing wrong?

Hint: An exponentially distributed random variable has positive support.
