Let $X$ be a random variable with uniform distribution on $[-1, 1]$. Find the CDF of random variable Y given by the following formula:
$Y = \left\{\begin{matrix}
-\frac{1}{2},& X < – \frac{1}{2}\\
X,& -\frac{1}{2} \leq X \leq \frac{1}{4}\\
\frac{1}{4}, & X > \frac{1}{4}
\end{matrix}\right.$
So I've found PDF and CDF of $X$:
$f_X(x) = \begin{cases}
\frac{1}{2}, & x \in [-1, 1]\\
0, & \text{otherwise}
\end{cases}$
$F_X(a) = \int_{-\infty}^{a} f_X(x) dx = \left\{\begin{matrix}
0, & a \leq -1\\
\frac{a+1}{2}, & a \in (-1, 1) \\
1, & a \geq 1
\end{matrix}\right.$
I tried to find Y's CDF by:
$F_Y(a) = P(Y \leq a)$
$= P(-\frac{1}{2} \leq a, X < – \frac{1}{2}) + P(X \leq a, – \frac{1}{2} \leq X \leq \frac{1}{4}) + P(\frac{1}{4} \leq a, X > -\frac{1}{4})$
But what should I do next? I'm finding such CDF for the first time and my notes say I need to consider a few different cases, but I have no clue what they should look like and how to do it. Any tips would be helpful.
Best Answer
Tip: You can find a CDF, but not a pdf, for all points on the support.
Notice that $Y$ will have a support of the interval $[-\tfrac 12;\tfrac 14]$, but will have a probability mass at the two end points, with a probability density in the interval between. It is a mixed distribution.
That is $~\mathsf P(Y{=}-\tfrac 12)~=~\mathsf P(-1{\leq}X{<}-\tfrac 12)~$ and $~\mathsf P(Y{=}\tfrac 14)~=~\mathsf P(\tfrac 14{<}X{\leq}1)~$.
So then
If you've peeked, notice the step discontinuities.