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 $$ p_{\left|\mathbf{X}\right|}\left(x\right) = \left(\begin{array}{c} 1\\ x \end{array}\right)\frac{3}{4}^{x}\frac{1}{4}^{1-x} = \begin{cases} \frac{3}{4} & \text{if }x=1\\ \frac{1}{4} & \text{if }x=0 \end{cases} $$