Find probability distribution function of a random process

Question

I am trying to figure out how random processes work and I am not sure if I get it right.

There's an example in my book for finding the probability density function and as an extra I am trying to find the probability distribution function.

I have a random sequence of 0 and 1. They both have the same probability to appear.

I also have two functions

$x_1(t) = sin (πt)$ for the 0's

$x_2(t) = -sin(πt)$ for the 1's

and I need to find the probability distribution function of two random variables for $t = 0$ and $t = 0.5$

So what I am doing is:

For $t = 0$:

$x_1(t) = 0$

$x_2(t) = 0$

so my random variable is $X_0=X(0)$ with a probability $P(X_0) = 1$

and the function should be:

$F_{x0}(x) = P(X_0\le\ x)= \begin{cases} 0 & -\infty \leq x\lt 0 \\ 1 & 0 \leq x\lt \infty \end{cases}$

For $t = 0.5$:

$x_1(t) = 1$

$x_2(t) = -1$

so my random variable is $X_{0.5}=X(0.5)$ with a probability $P(X_{0.5} = 1) = {1 \over 2}$ and $P(X_{0.5} = -1) = {1 \over 2}$

and the function should be:

$F_{x0.5}(x) = P(X_{0.5}\le\ x)= \begin{cases} 0 & -\infty \leq x\lt -1 \\ 1 \over 2\ & -1 \leq x \lt 0\\ 0 & 0 \leq x \lt 1\\ 1 & 1 \leq x\lt \infty \end{cases}$

or since we need to sum up, it should be like this?

$F_{x0.5}(x) = P(X_{0.5}\le\ x)= \begin{cases} 0 & -\infty \leq x\lt -1 \\ 1 \over 2\ & -1 \leq x \lt 0\\ 1 & 1 \leq x\lt \infty \end{cases}$

So, is what I am doing correct?

Thanks