Cumulative distribution function with 3 variables

Question

Let $X$ be the random variable whose cumulative distribution function is $$ F_X (x) = \begin{cases} 0, & \text{for} \space x\lt 0 \\ \frac{1}{2}, & \text{for} \space 0\le x\le 1 \\ 1, & \text{for} \space x\gt 1 \\ \end{cases}. $$ Let $Y$ be a random variable independent of $X$ and uniformly distributed over the interval $(0,1)$. Define the random variable $Z$ as $$ Z = \begin {cases} X, & \text{if} \space X\le \frac{1}{2} \\ Y, & \text{if} \space X\gt \frac{1}{2} \\ \end{cases} $$ Determine $\mathbb{P} (Z\le \frac{1}{5})$.

I believe that $X$ only takes the discrete values $0$ and $1$ with equal probability, but I'm not entirely sure. By intuition, I think that the answer is $\frac{1}{2}$. I'm unsure about this question, so any advice would be appreciated.

Answer 1

You should define $F(1)$ as $1$ instead of $\frac 1 2$.

$$P(Z \leq \frac 1 5)$$ $$=P(X \leq \frac 1 5)+P(Y \leq \frac 1 5, X>\frac 1 2)$$ $$=P(X \leq \frac 1 5)+P(Y \leq \frac 1 5)P( X>\frac 1 2)$$

$$=\frac 1 2 +\frac 1 5 (1-\frac 1 2 )=\frac 3 5.$$

Answer 2

I believe that $X$ only takes the discrete values $0$ and $1$ with equal probability, but I'm not entirely sure.

Yes, that is what the cumulative distribution function is saying: as it increases suddenly at $0$ and $1$, and remains constant elsewhere.  Thus $X$ has the probability mass function of:

$\qquad\mathsf P(X=x)=\begin{cases}1/2 &:& x=0~\text{or}~ x=1\\0&:&\text{elsewhere}\end{cases}$

So you should see that when $X=0$ then $Z=0$, or when $X=1$ then $Z=Y$, so: $$\mathsf P(Z\leq 1/5)=\mathsf P(X = 0)+\mathsf P(X=1\cap Y\leq 1/5)$$