Probability mass function

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 ${\displaystyle p:\mathbb {\mathbb {R} } }$ ${\displaystyle \rightarrow [0,1]}$ defined by

for ${\displaystyle -\infty <x<\infty }$,[2] where ${\displaystyle P}$ is a probability measure. ${\displaystyle p_{X}(x)}$ can also be simplified as ${\displaystyle p(x)}$.[3]

The probabilities associated with all possible values must be positive and sum up to 1. For all other values, the probabilities need to be 0.

${\displaystyle \sum p_{X}(x_{i})=1}$

${\displaystyle p(x_{i})>0}$

${\displaystyle p(x)=0}$ for all other x

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

A probability mass function of a discrete random variable ${\displaystyle X}$ can be seen as a special case of two more general measure theoretic constructions: the distribution of ${\displaystyle X}$ and the probability density function of ${\displaystyle X}$ with respect to the counting measure. We make this more precise below.

Suppose that ${\displaystyle (A,{\mathcal {A}},P)}$ is a probability space and that ${\displaystyle (B,{\mathcal {B}})}$ is a measurable space whose underlying σ-algebra is discrete, so in particular contains singleton sets of ${\displaystyle B}$. In this setting, a random variable ${\displaystyle X\colon A\to B}$ is discrete provided its image is countable. The pushforward measure ${\displaystyle X_{*}(P)}$—called a distribution of ${\displaystyle X}$ in this context—is a probability measure on ${\displaystyle B}$ whose restriction to singleton sets induces a probability mass function ${\displaystyle f_{X}\colon B\to \mathbb {R} }$ since ${\displaystyle f_{X}(b)=P(X^{-1}(b))=[X_{*}(P)](\{b\})}$ for each ${\displaystyle b\in B}$.

Now suppose that ${\displaystyle (B,{\mathcal {B}},\mu )}$ is a measure space equipped with the counting measure μ. The probability density function ${\displaystyle f}$ of ${\displaystyle X}$ with respect to the counting measure, if it exists, is the Radon–Nikodym derivative of the pushforward measure of ${\displaystyle X}$ (with respect to the counting measure), so ${\displaystyle f=dX_{*}P/d\mu }$ and ${\displaystyle f}$ is a function from ${\displaystyle B}$ to the non-negative reals. As a consequence, for any ${\displaystyle b\in B}$ we have

${\displaystyle P(X=b)=P(X^{-1}(\{b\})):=\int _{X^{-1}(\{b\})}dP=}$${\displaystyle \int _{\{b\}}fd\mu =f(b),}$

demonstrating that ${\displaystyle f}$ is in fact a probability mass function.

When there is a natural order among the potential outcomes ${\displaystyle x}$, it may be convenient to assign numerical values to them (or n-tuples in case of a discrete multivariate random variable) and to consider also values not in the image of ${\displaystyle X}$. That is, ${\displaystyle f_{X}}$ may be defined for all real numbers and ${\displaystyle f_{X}(x)=0}$ for all ${\displaystyle x\notin X(S)}$ as shown in the figure.

The image of ${\displaystyle X}$ has a countable subset on which the probability mass function ${\displaystyle f_{X}(x)}$ is one. Consequently, the probability mass function is zero for all but a countable number of values of ${\displaystyle x}$.

The discontinuity of probability mass functions is related to the fact that the cumulative distribution function of a discrete random variable is also discontinuous. If ${\displaystyle X}$ is a discrete random variable, then ${\displaystyle P(X=x)=1}$ means that the casual event ${\displaystyle (X=x)}$ is certain (it is true in the 100% of the occurrencies); on the contrary, ${\displaystyle P(X=x)=0}$ means that the casual event ${\displaystyle (X=x)}$ is always impossible. This statement isn't true for a continuous random variable ${\displaystyle X}$, for which ${\displaystyle P(X=x)=0}$ for any possible ${\displaystyle x}$: in fact, by definition, a continuous random variable can have an infinite set of possible values and thus the probability it has a single particular value x is equal to ${\displaystyle {\frac {1}{\infty }}=0}$. Discretization is the process of converting a continuous random variable into a discrete one.

Two or more discrete random variables have a joint probability mass function, which gives the probability of each possible combination of realizations for the random variables.

• Johnson, N. L.; Kotz, S.; Kemp, A. (1993). Univariate Discrete Distributions (2nd ed.). Wiley. p. 36. ISBN 0-471-54897-9.