Use a probability distribution

Status

Variants

Evaluate a probability distribution

Compute densities, probabilities or samples from a named probability distribution.

What other tools call it

Meridian's classification; an independent blind rating agreed on 86.2% of 312 rows (exact class, version 2).

Background

When to use it

The quantile function of a probability distribution is the inverse of its cumulative distribution function.^[2] A statistical application may need key percentage points of a distribution, such as its 2.5% and 97.5% levels, to assess the statistical significance of an observation whose distribution is known.^[2] The quantile of the standard normal distribution at 0.975 is 1.96.^[3] Quantile functions are used in both statistical applications and Monte Carlo methods.^[2]

Unlike a probability, a probability density function can take on values greater than one.^[4] The probability for a continuous random variable to take on any particular value is zero.^[4] The probability of a continuous random variable falling within a range of values is given by the integral of its probability density function over that range.^[4] Treat a density value as a relative likelihood to compare with other density values, and turn to the cumulative distribution function when the question is the chance of a range. (Meridian's view)

The method

The cumulative distribution function of a real-valued random variable X, evaluated at x, is the probability that X will take a value less than or equal to x.^[5] For a continuous distribution, the cumulative distribution function at a value is the area under the probability density function up to that value.^[1] Probability mass functions are used for discrete distributions, while probability density functions are used for many continuous distributions.^[1] The probability density function of a continuous random variable can be determined from its cumulative distribution function by differentiating.^[5]

Inverse transform sampling, also known as inversion sampling, is a basic method for generating sample numbers at random from any probability distribution given its cumulative distribution function.^[6] Inverse transform sampling takes uniform samples of a number between 0 and 1, interpreted as a probability.^[6] Computationally, inverse transform sampling involves computing the quantile function of the distribution.^[6] For the normal distribution, the lack of an analytical expression for the quantile function means that other methods, such as the Box–Muller transform, may be preferred computationally.^[6] Inversion sampling is now the default method for sampling from a normal distribution in the statistical package R.^[6]

In mlpack, the GaussianDistribution, GammaDistribution and LaplaceDistribution classes each compute probabilities of points, sample from the distribution and fit the distribution to observations.^[7]

Compared with the alternatives

Parametric statistics assumes that the distribution the observed data was drawn from can be described by a finite set of unknown parameters.^[8] Nonparametric statistics does not assume explicit finite-parametric mathematical forms for distributions when modelling data.^[8] Most well-known statistical methods are parametric.^[8] Nonparametric methods make fewer assumptions, so their applicability is much more general than that of the corresponding parametric methods.^[9] When the form of a distribution is in doubt, an estimate from the sample is a gentler place to start than a named distribution chosen on faith. (Meridian's view)

Sources

1. [1]Probability distributionsecondary source
2. [2]Quantile functionsecondary source
3. [3]Normal distributionsecondary source
4. [4]Probability density functionsecondary source
5. [5]Cumulative distribution functionsecondary source
6. [6]Inverse transform samplingsecondary source
7. [7]mlpack documentationdocumentation
8. [8]Parametric statisticssecondary source
9. [9]Nonparametric statisticssecondary source