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Formal definition

Probability mass function is the probability distribution of a discrete random variable, and provides the possible values and their associated probabilities. It is the function p: R → {\displaystyle p:\mathbb {R} \to } defined by

p X (x) = P (X = x) {\displaystyle p_{X}(x)=P(X=x)}

for − ∞ < x < ∞ {\displaystyle -\infty <x<\infty }, where P {\displaystyle P} is a probability measure. p X (x) {\displaystyle p_{X}(x)} can also be simplified as p (x) {\displaystyle p(x)}.

The probabilities associated with all (hypothetical) values must be non-negative and sum up to 1,

and

Thinking of probability as mass helps to avoid mistakes since the physical mass is conserved as is the total probability for all hypothetical outcomes x {\displaystyle x}.