Probability mass function - Wikipedia (FazBrowse)

The graph of a PMF. All the values of this function must be non-negative and sum up to 1.

In probability and statistics, a probability mass function (PMF), discrete probability density function, probability function, or frequency function is a function that gives the probability that a discrete random variable is exactly equal to some value. The PMF is often the primary means of defining a discrete probability distribution, and such functions exist for either scalar or multivariate random variables whose domain is discrete.

A PMF differs from a continuous probability density function (PDF) in that the latter is associated with continuous rather than discrete random variables. A continuous PDF must be integrated over an interval to yield a probability.

The value of the random variable having the largest probability mass is called the mode.

P is a probability measure. ∑xpX(x)=1 and X can be seen as a special case of two more general measure theoretic constructions: the distribution of (A,𝒜,P) is a probability space and that X:A→B is discrete provided its image is countable. The pushforward measure B whose restriction to singleton sets induces the PMF (as mentioned in the previous section) b∈B.

Now suppose that f of f=dX∗P/dμ and b∈B we have 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 fX(x)=0 for all fX(x) is one. Consequently, the PMF is zero for all but a countable number of values of P(X=x)=1 means that the casual event (X=x) is always impossible. This statement isn't true for a continuous random variable x. Discretization is the process of converting a continuous random variable into a discrete one.

Examples

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Finite

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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. X is the random variable defined on (nk)pk(1−p)n−k. The PMF of a fair die. All the numbers on the die have an equal chance of appearing on top when the die stops rolling.
An example is tossing a coin until the first "heads" appears.

Other distributions that can be modeled using a PMF 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 PMF, 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.

Stewart, William J. (2011). Probability, Markov Chains, Queues, and Simulation: The Mathematical Basis of Performance Modeling. Princeton University Press. p. 105. ISBN 978-1-4008-3281-1.

1 2 Johnson, N. L.; Kotz, S.; Kemp, A. (1993). Univariate Discrete Distributions (2nd ed.). Wiley. p. 36. ISBN 0-471-54897-9.

Theory of probability distributions