Probability mass function

Measure theoretic formulation

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 ${\displaystyle X:A\to B}$ is discrete provided its image is countable. The pushforward measure ${\displaystyle X_{\ast }(P)}$—called the distribution of ${\displaystyle X}$ in this context—is a probability measure on ${\displaystyle B}$ whose restriction to singleton sets induces the probability mass function (as mentioned in the previous section) ${\displaystyle f_X:B\to \mathbb{R}}$ since ${\displaystyle f_X(b)=P(X^{-1}(b))=P(X=b)}$ for each ${\displaystyle b\in B}$.

Now suppose that ${\displaystyle (B,\mathcal{B},\mu )}$ is a measure space equipped with the counting measure ${\displaystyle \mu }$. 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 ${\displaystyle f=d{X}_{\ast }P/d\mu }$ 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 $${\displaystyle P(X=b)=P(X^{-1}(b))=X_{\ast }(P)(b)={\int }_{b}fd\mu =f(b),}$$

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 100% of the occurrences); 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 ${\displaystyle x}$. Discretization is the process of converting a continuous random variable into a discrete one.

Examples

Finite

There are three major distributions associated, the Bernoulli distribution, the binomial distribution and the geometric distribution.

• Bernoulli distribution: ber(p) , is used to model an experiment with only two possible outcomes. The two outcomes are often encoded as 1 and 0. $${\displaystyle p_X(x)=\{\begin{array}{ll}p,& \text{if }x\text{ is 1}\\ 1-p,& \text{if }x\text{ is 0}\end{array}}$$ An example of the Bernoulli distribution is tossing a coin. Suppose that ${\displaystyle S}$ is the sample space of all outcomes of a single toss of a fair coin, and ${\displaystyle X}$ is the random variable defined on ${\displaystyle S}$ assigning 0 to the category "tails" and 1 to the category "heads". Since the coin is fair, the probability mass function is $${\displaystyle p_X(x)=\{\begin{array}{ll}\frac{1}{2},& x=0,\\ \frac{1}{2},& x=1,\\ 0,& x\notin \{0,1\}.\end{array}}$$
• Binomial distribution, models the number of successes when someone draws n times with replacement. Each draw or experiment is independent, with two possible outcomes. The associated probability mass function is $(\genfrac{}{}{0ex}{}{n}{k})p^k(1-p{)}^{n-k}$. An example of the binomial distribution is the probability of getting exactly one 6 when someone rolls a fair die three times.
• Geometric distribution describes the number of trials needed to get one success. Its probability mass function is $p_X(k)=(1-p{)}^{k-1}p$.An example is tossing a coin until the first "heads" appears. ${\displaystyle p}$ denotes the probability of the outcome "heads", and ${\displaystyle k}$ denotes the number of necessary coin tosses. Other distributions that can be modeled using a probability mass function are the categorical distribution (also known as the generalized Bernoulli distribution) and the multinomial distribution.
• If the discrete distribution has two or more categories one of which may occur, whether or not these categories have a natural ordering, when there is only a single trial (draw) this is a categorical distribution.
• An example of a multivariate discrete distribution, and of its probability mass function, is provided by the multinomial distribution. Here the multiple random variables are the numbers of successes in each of the categories after a given number of trials, and each non-zero probability mass gives the probability of a certain combination of numbers of successes in the various categories.

Infinite

The following exponentially declining distribution is an example of a distribution with an infinite number of possible outcomes—all the positive integers: $${\displaystyle \text{Pr}(X=i)=\frac{1}{2^i}\text{for }i=1,2,3,\dots }$$ Despite the infinite number of possible outcomes, the total probability mass is 1/2 + 1/4 + 1/8 + ⋯ = 1, satisfying the unit total probability requirement for a probability distribution.

Multivariate case

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.