Probability Mass Function of Function of Discrete Random Variable

Theorem
Let $$X$$ be a discrete random variable.

Let $$Y = g \left({X}\right)$$, where $$g: \R \to \R$$ is a real function.

Then the probability mass function of $$Y$$ is given by:
 * $$p_Y \left({y}\right) = \sum_{x \in g^{-1} \left({y}\right)} \Pr \left({X = x}\right)$$

Proof
By Function of Discrete Random Variable‎ we have that $$Y$$ is itself a discrete random variable.

Thus:

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