Proving that a discrete stochastic variable is binomial distributed. Given a discrete stochastic variables, with the probability function;
$$
p_{X}\left(x\right)=\left\{
\begin{array}{cc}
\frac{1}{4} & \text{if } x = -1 \\
\frac{1}{4} & \text{if } x = 0 \\
\frac{1}{2} & \text{if } x = 1 \\
0 & \text{otherwise}
\end{array}
\right.
$$
How do I prove that;
$$|X| \sim b\left(1, \frac{3}{4}\right)$$
What I've done so far: Found $P(|X| = 1) = \frac{3}{4}$, and then I've unsuccessfully tried to derive the probability function (above) from the binomial probability function (below);
$$
p\left(x\right) = \left(
\begin{array}{c}
n \\ x
\end{array}\right) \pi^{x} \left(1 - \pi\right)^{n-x}
$$
However without any luck, as $|X| \neq X$.
So my question is somewhat simple; How would I go about proving that a discrete stochastic variable is binomial distributed?

You already did all of the work. You showed that
$$
p_{\left|\mathbf{X}\right|}\left(x\right)
=
\left(\begin{array}{c}
1\\
x
\end{array}\right)\frac{3}{4}^{x}\frac{1}{4}^{1-x}
=
\begin{cases}
\frac{3}{4} & \text{if }x=1\\
\frac{1}{4} & \text{if }x=0
\end{cases}
$$
