Probability distribution

In probability theory and statistics, a probability distribution describes how probabilities are assigned to the possible results of a random phenomenon, more precisely, to events, which are sets of possible outcomes of a probabilistic experiment. Informally, a probability distribution tells us how likely different results are. Formally, it is a probability measure: a function that assigns probabilities to events in a way that satisfies the axioms of probability.

Probability distributions are closely linked to random variables. A random variable is a function that assigns a value to each outcome of a probabilistic experiment; it induces a probability distribution on the set of values it can take. For example, the result of a coin toss can be represented by a random variable X that equals 1 for heads and 0 for tails. If the coin is fair, this distribution assigns probability 1/2 to X = 1 and probability 1/2 to X = 0. Viewed as a probability measure, the distribution of X assigns ℙ(X ∈ A) to each set A ⊆ {0,1}; for a fair coin, ℙ(X ∈ {1}) = ℙ(X ∈ {0}) = 1/2, ℙ(X ∈ {0,1}) = 1, and ℙ(X ∈ ∅) = 0.

In practice, probability distributions are often described by functions such as cumulative distribution functions, probability mass functions, or probability density functions. Which description is used depends on the nature of the distribution: probability mass functions are used for discrete distributions, while probability density functions are used for many continuous distributions.

Probability distributions that occur frequently or have special theoretical importance are often given specific names; examples are collected in the list of probability distributions.

A probability distribution is a mathematical description of the probabilities of events, i.e. subsets of the sample space. The sample space, often represented in notation by \(\ \Omega\ ,\) is the set of all possible outcomes of a random phenomenon being observed. The sample space may be any set of numbers, vectors, labels, or whatever else. For example, the sample space of a coin flip could be Ω = {"heads", "tails"} , whereas for a die roll, it could be Ω = {1, 2, 3, 4, 5, 6} .

To define probability distributions for the specific case of random variables (so that the sample space can mapped to a measurable space, for example the real numbers), it is common to distinguish between discrete and continuous random variables. In the discrete case, it is sufficient to specify a probability mass function \(p\) assigning a probability to each possible outcome (e.g. when throwing a fair die, each of the six digits "1" to "6", corresponding to the number of dots on the die, has probability \(\tfrac{1}{6}\) of being on top when it lands). The probability of an event is then defined to be the sum of the probabilities of all outcomes that satisfy the event; for example, the probability of the event "the die rolls an even value" is \[P(2) + P(4) + P(6) = \frac{1}{6} + \frac{1}{6} + \frac{1}{6} = \frac{1}{2}\] In contrast, when a random variable takes values from a continuum, then unless the probability density function has any infinitely-dense peaks, any individual outcome has probability zero. For such continuous random variables, only events that include infinitely many outcomes, such as intervals, have probability greater than 0.

For example, consider measuring the weight of a piece of ham in the supermarket, and assume the scale can provide arbitrarily many digits of precision. Then, the probability that it weighs exactly 500 g must be zero because no matter how high the level of precision chosen, it cannot be assumed that there are no non-zero digits after those output by the scale. However, for the same use case, it is possible to meet quality control requirements such as that a package of "500 g" of ham must weigh between 490 g and 510 g. This is possible because this measurement does not require infinite precision from the underlying equipment, and it provides some tolerance for variability in physical objects and processes.

Continuous probability distributions can be described by means of the cumulative distribution function, which describes the probability that the random variable is no larger than a given value (i.e., P(X ≤ x) for some x. The cumulative distribution function is the area under the probability density function from -∞ to x, as shown in figure 1.

Condensed: the full section is in Wikipedia.

Let \((\Omega, \mathcal{F}, P)\) be a probability space, \((E, \mathcal{E})\) be a measurable space, and \(X : \Omega \to E\) be a \((E, \mathcal{E})\)-valued random variable. Then the probability distribution of \(X\) is the pushforward measure of the probability measure \(P\) onto \((E, \mathcal{E})\) induced by \(X\). Explicitly, this pushforward measure on \((E, \mathcal{E})\) is given by \[X_{*} (P) (B) = P \left( X^{-1} (B) \right)\] for \(B \in \mathcal{E}.\)

Any probability distribution is a probability measure on \((E, \mathcal{E})\) (in general different from \(P\), unless \(X\) happens to be the identity map).

A probability distribution can be described in various forms, such as by a probability mass function or a cumulative distribution function. One of the most general descriptions, which applies for absolutely continuous and discrete variables, is by means of a probability function \(P \colon \mathcal{A} \to \Reals\) whose input space \(\mathcal{A}\) is a σ-algebra, and gives a real number probability as its output, particularly, a number in \([0,1] \subseteq \Reals\).

The probability function \(P\) can take as argument subsets of the sample space itself, as in the coin toss example, where the function \(P\) was defined so that P(heads) = 0.5 and P(tails) = 0.5. However, because of the widespread use of random variables, which transform the sample space into a set of numbers (e.g., \(\R\), \(\N\)), it is more common to study probability distributions whose argument are subsets of these particular kinds of sets (number sets), and all probability distributions discussed in this article are of this type. It is common to denote as \(P(X \in E)\) the probability that a certain value of the variable \(X\) belongs to a certain event \(E\).

The above probability function only characterizes a probability distribution if it satisfies all the Kolmogorov axioms, that is:

1. \(P(X \in E) \ge 0 \; \forall E \in \mathcal{A}\), so the probability is non-negative
2. \(P(X \in E) \le 1 \; \forall E \in \mathcal{A}\), so no probability exceeds \(1\)
3. \(P(X \in \bigcup_{i} E_i ) = \sum_i P(X \in E_i)\) for any countable disjoint family of sets \(\{ E_i \}\)

The concept of probability function is made more rigorous by defining it as the element of a probability space \((X, \mathcal{A}, P)\), where \(X\) is the set of possible outcomes, \(\mathcal{A}\) is the set of all subsets \(E \subset X\) whose probability can be measured, and \(P\) is the probability function, or probability measure, that assigns a probability to each of these measurable subsets \(E \in \mathcal{A}\).

Probability distributions usually belong to one of two classes.

Condensed: the full section is in Wikipedia.

• Random variable: takes values from a sample space; probabilities describe which values and set of values are more likely taken.
• Event: set of possible values (outcomes) of a random variable that occurs with a certain probability.
• Probability function or probability measure: describes the probability \(P(X \in E)\) that the event \(E,\) occurs.
• Cumulative distribution function: function evaluating the probability that \(X\) will take a value less than or equal to \(x\) for a random variable (only for real-valued random variables).
• Quantile function: the inverse of the cumulative distribution function. Gives \(x\) such that, with probability \(q\), \(X\) will not exceed \(x\).

• Discrete probability distribution: for many random variables with finitely or countably infinitely many values.
• Probability mass function (pmf): function that gives the probability that a discrete random variable is equal to some value.
• Frequency distribution: a table that displays the frequency of various outcomes in a sample.
• Relative frequency distribution: a frequency distribution where each value has been divided (normalized) by a number of outcomes in a sample (i.e. sample size).
• Categorical distribution: for discrete random variables with a finite set of values.

• Absolutely continuous probability distribution: for many random variables with uncountably many values.
• Probability density function (pdf) or probability density: function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) can be interpreted as providing a relative likelihood that the value of the random variable would equal that sample.

• Support: the set of values x such that the random variable as a positive probability of falling in every open neighborhood of x.
• Tail: the regions close to the bounds of the random variable, if the pmf or pdf are relatively low therein. Usually has the form \(X > a\), \(X < b\) or a union thereof.
• Expected value or mean: the weighted average of the possible values, using their probabilities as their weights; or the continuous analog thereof.
• Median: the value such that the set of values less than the median, and the set greater than the median, each have probabilities no greater than one-half.
• Mode: for a discrete random variable, the value with highest probability; for an absolutely continuous random variable, a location at which the probability density function has a local peak.
• Quantile: the q-quantile is the value \(x\) such that \(P(X < x) = q\).
• Variance: the second moment of the random variable about its mean; an important measure of the dispersion of the distribution.
• Standard deviation: the square root of the variance, and hence another measure of dispersion.
• Symmetry: a property of some distributions in which the portion of the distribution to the left of a specific value (usually the median) is a mirror image of the portion to its right.
• Skewness: a measure of the extent to which a pmf or pdf "leans" to one side of its mean. The third standardized moment of the distribution.
• Kurtosis: a measure of the "fatness" of the tails of a pmf or pdf. The fourth standardized moment of the distribution.

In the special case of a real-valued random variable, the probability distribution can equivalently be represented by a cumulative distribution function instead of a probability measure. The cumulative distribution function of a random variable \(X\) with regard to a probability distribution \(p\) is defined as \[F(x) = P(X \leq x).\]

The cumulative distribution function of any real-valued random variable has the properties:

• \(F(x)\) is non-decreasing;
• \(F(x)\) is right-continuous;
• \(0 \le F(x) \le 1\);
• \(\lim_{x \to -\infty} F(x) = 0\) and \(\lim_{x \to \infty} F(x) = 1\); and
• \(\Pr(a < X \le b) = F(b) - F(a)\).

Conversely, any function \(F:\mathbb{R}\to\mathbb{R}\) that satisfies the first four of the properties above is the cumulative distribution function of some probability distribution on the real numbers.

Any probability distribution can be decomposed as the mixture of a discrete, an absolutely continuous and a singular continuous distribution, and thus any cumulative distribution function admits a decomposition as the convex sum of the three according cumulative distribution functions.

A discrete probability distribution is the probability distribution of a random variable that can take on only a countable number of values (almost surely) which means that the probability of any event \(E\) can be expressed as a (finite or countably infinite) sum: \[P(X\in E) = \sum_{\omega\in A \cap E} P(X = \omega),\] where \(A\) is a countable set with \(P(X \in A) = 1\). Thus the discrete random variables (i.e. random variables whose probability distribution is discrete) are exactly those with a probability mass function \(p(x) = P(X=x)\). In the case where the range of values is countably infinite, these values have to decline to zero fast enough for the probabilities to add up to 1. For example, if \(p(n) = \tfrac{1}{2^n}\) for \(n = 1, 2, ...\), the sum of probabilities would be \(1/2 + 1/4 + 1/8 + \dots = 1\).

Well-known discrete probability distributions used in statistical modeling include the Poisson distribution, the Bernoulli distribution, the binomial distribution, the geometric distribution, the negative binomial distribution and categorical distribution. When a sample (a set of observations) is drawn from a larger population, the sample points have an empirical distribution that is discrete, and which provides information about the population distribution. Additionally, the discrete uniform distribution is commonly used in computer programs that make equal-probability random selections between a number of choices.

A real-valued discrete random variable can equivalently be defined as a random variable whose cumulative distribution function increases only by jump discontinuities, that is, its cdf increases only where it "jumps" to a higher value, and is constant in intervals without jumps. The points where jumps occur are precisely the values which the random variable may take. Thus the cumulative distribution function has the form \[F(x) = P(X \leq x) = \sum_{\omega \leq x} p(\omega).\] The points where the cdf jumps always form a countable set; this may be any countable set and thus may even be dense in the real numbers.

A discrete probability distribution is often represented with Dirac measures, also called one-point distributions (see below), the probability distributions of deterministic random variables. For any outcome \(\omega\), let \(\delta_\omega\) be the Dirac measure concentrated at \(\omega\). Given a discrete probability distribution, there is a countable set \(A\) with \(P(X \in A) = 1\) and a probability mass function \(p\). If \(E\) is any event, then \[P(X \in E) = \sum_{\omega \in A} p(\omega) \delta_\omega(E),\] or in short, \[P_X = \sum_{\omega \in A} p(\omega) \delta_\omega.\]

Similarly, discrete distributions can be represented with the Dirac delta function as a generalized probability density function \(f\), where \[f(x) = \sum_{\omega \in A} p(\omega) \delta(x - \omega),\] which means \[P(X \in E) = \int_E f(x) \, dx = \sum_{\omega \in A} p(\omega) \int_E \delta(x - \omega) = \sum_{\omega \in A \cap E} p(\omega)\] for any event \(E.\)

For a discrete random variable \(X\), let \(u_0, u_1, \dots\) be the values it can take with non-zero probability. Denote \[\Omega_i=X^{-1}(u_i)= \{\omega: X(\omega)=u_i\},\, i=0, 1, 2, \dots\] These are disjoint sets, and for such sets \[P\left(\bigcup_i \Omega_i\right)=\sum_i P(\Omega_i)=\sum_i P(X=u_i)=1.\] It follows that the probability that \(X\) takes any value except for \(u_0, u_1, \dots\) is zero, and thus one can write \(X\) as \[X(\omega)=\sum_i u_i 1_{\Omega_i}(\omega)\] except on a set of probability zero, where \(1_A\) is the indicator function of \(A\). This may serve as an alternative definition of discrete random variables.

A special case is the discrete distribution of a random variable that can take on only one fixed value, in other words, a Dirac measure. Expressed formally, the random variable \(X\) has a one-point distribution if it has a possible outcome \(x\) such that \(P(X{=}x)=1.\) All other possible outcomes then have probability 0. Its cumulative distribution function jumps immediately from 0 before \(x\) to 1 at \(x\). It is closely related to a deterministic distribution, which cannot take on any other value, while a one-point distribution can take other values, though only with probability 0. For most practical purposes the two notions are equivalent.

An absolutely continuous probability distribution is a probability distribution on the real numbers with uncountably many possible values, such as a whole interval in the real line, and where the probability of any event can be expressed as an integral. More precisely, a real random variable \(X\) has an absolutely continuous probability distribution if there is a function \(f: \Reals \to [0, \infty]\) such that for each interval \(I = [a,b] \subset \mathbb{R}\) the probability of \(X\) belonging to \(I\) is given by the integral of \(f\) over \(I\): \[P\left(a \le X \le b \right) = \int_a^b f(x) \, dx .\] This is the definition of a probability density function, so that absolutely continuous probability distributions are exactly those with a probability density function. In particular, the probability for \(X\) to take any single value \(a\) (that is, \(a \le X \le a\)) is zero, because an integral with coinciding upper and lower limits is always equal to zero. If the interval \([a,b]\) is replaced by any measurable set \(A\), the according equality still holds: \[P(X \in A) = \int_A f(x) \, dx .\]

An absolutely continuous random variable is a random variable whose probability distribution is absolutely continuous.

There are many examples of absolutely continuous probability distributions: normal, uniform, chi-squared, and others.