Probability mass function explained
In probability and statistics, a probability mass function (sometimes called probability function or frequency function[1]) is a function that gives the probability that a discrete random variable is exactly equal to some value.[2] Sometimes it is also known as the discrete probability density function. The probability mass function 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 probability mass function differs from a probability density function (PDF) in that the latter is associated with continuous rather than discrete random variables. A PDF must be integrated over an interval to yield a probability.[3]
The value of the random variable having the largest probability mass is called the mode.
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
defined by
for
, where
is a
probability measure.
can also be simplified as
.
[4] 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
.
Measure theoretic formulation
A probability mass function of a discrete random variable
can be seen as a special case of two more general measure theoretic constructions: the
distribution of
and the
probability density function of
with respect to the
counting measure. We make this more precise below.
Suppose that
is a
probability spaceand that
is a measurable space whose underlying
σ-algebra is discrete, so in particular contains singleton sets of
. In this setting, a random variable
is discrete provided its image is countable.The
pushforward measure
—called the distribution of
in this context—is a probability measure on
whose restriction to singleton sets induces the probability mass function (as mentioned in the previous section)
since
for each
.
Now suppose that
is a
measure space equipped with the counting measure
. The probability density function
of
with respect to the counting measure, if it exists, is the Radon–Nikodym derivative of the pushforward measure of
(with respect to the counting measure), so
and
is a function from
to the non-negative reals. As a consequence, for any
we have
demonstrating that
is in fact a probability mass function.
When there is a natural order among the potential outcomes
, 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
. That is,
may be defined for all
real numbers and
for all
as shown in the figure.
The image of
has a
countable subset on which the probability mass function
is one. Consequently, the probability mass function is zero for all but a countable number of values of
.
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
is a discrete random variable, then
means that the casual event
is certain (it is true in 100% of the occurrences); on the contrary,
means that the casual event
is always impossible. This statement isn't true for a continuous random variable
, for which
for any possible
.
Discretization is the process of converting a continuous random variable into a discrete one.
Examples
See main article: Bernoulli distribution, Binomial distribution and Geometric distribution.
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.
p, & \textx\text \\1-p, & \textx\text\end An example of the Bernoulli distribution is tossing a coin. Suppose that
is the sample space of all outcomes of a single toss of a fair coin, and
is the random variable defined on
assigning 0 to the category "tails" and 1 to the category "heads". Since the coin is fair, the probability mass function is
- 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 . 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 .An example is tossing a coin until the first "heads" appears.
denotes the probability of the outcome "heads", and
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: 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
See main article: Joint probability distribution.
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.
Further reading
Notes and References
- https://online.stat.psu.edu/stat414/lesson/7/7.2 7.2 - Probability Mass Functions | STAT 414 - PennState - Eberly College of Science
- Book: Stewart, William J.. Probability, Markov Chains, Queues, and Simulation: The Mathematical Basis of Performance Modeling. Princeton University Press. 2011. 978-1-4008-3281-1. 105.
- Book: A modern introduction to probability and statistics : understanding why and how. 2005. Springer. Dekking, Michel, 1946-. 978-1-85233-896-1. London. 262680588.
- Book: Rao, Singiresu S.. Engineering optimization : theory and practice. 1996. Wiley. 0-471-55034-5. 3rd. New York. 62080932.