Methods and formulas for Probability Distributions

For a number p in the closed interval [0,1], the inverse cumulative distribution function (ICDF) of a random variable X determines, where possible, a value x such that the probability of X ≤ x is greater than or equal to p.

The ICDF for continuous distributions

The ICDF is the value that is associated with an area under the probability density function. The ICDF is the reverse of the cumulative distribution function (CDF), which is the area that is associated with a value.

For all continuous distributions, the ICDF exists and is unique if 0 < p < 1.

• When the probability density function (PDF) is positive for the entire real number line (for example, the normal PDF), the ICDF is not defined for either p = 0 or p = 1.
• When the PDF is positive for all values that are greater than some value (for example, the chi-square PDF), the ICDF is defined for p = 0 but not for p = 1.
• When the PDF is positive only on an interval (for example, the uniform PDF), the ICDF is defined for p = 0 and p = 1.
• When the ICDF is not defined, Minitab returns a missing value (*) for the result.

The ICDF for discrete distributions

The ICDF is more complicated for discrete distributions than it is for continuous distributions. When you calculate the CDF for a binomial with, for example, n = 5 and p = 0.4, there is no value x such that the CDF is 0.5. For x = 1, the CDF is 0.3370. For x = 2, the CDF increases to 0.6826.

When the ICDF is displayed (that is, the results are not stored), both values of x are displayed. When the ICDF is stored, the larger of the two values is stored.

The beta distribution is often used to represent processes with natural lower and upper limits.

Formula

The probability density function (PDF) is:

Notation

When a = 0, b = 1,

the PDF is:

The binomial distribution is used to represent the number of events that occurs within n independent trials. Possible values are integers from zero to n.

Formula

mean = np

variance = np(1 – p)

The probability mass function (PMF) is:

Where equals .

In general, you can calculate k! as

Notation

The Cauchy distribution is symmetric around zero, but the tails approach zero less quickly than do those of the normal distribution.

Formula

The probability density function (PDF) is:

Notation

If X has a standard normal distribution, X² has a chi-square distribution with one degree of freedom, allowing it to be a commonly used sampling distribution.

The sum of n independent X² variables (where X has a standard normal distribution) has a chi-square distribution with n degrees of freedom. The shape of the chi-square distribution depends on the number of degrees of freedom.

Formula

The probability density function (PDF) is:

mean = v

variance = 2v

Notation

A discrete distribution is one that you define yourself. For example, suppose you are interested in a distribution made up of three values −1, 0, 1, with probabilities of 0.2, 0.5, and 0.3, respectively. If you enter the values into columns of a worksheet, then you can use these columns to generate random data or to calculate probabilities.

The exponential distribution can be used to model time between failures, such as when units have a constant, instantaneous rate of failure (hazard function). The exponential distribution is a special case of the Weibull distribution and the gamma distribution.

Formula

The probability density function (PDF) is:

The cumulative distribution function (CDF) is:

mean = θ + λ

variance = θ²

Notation

The F-distribution is also known as the variance-ratio distribution and has two types of degrees of freedom: numerator degrees of freedom and denominator degrees of freedom. It is the distribution of the ratio of two independent random variables with chi-square distributions, each divided by its degrees of freedom.

Formula

The probability density function (PDF) is:

Notation

The gamma distribution is often used to model positively skewed data.

Formula

The probability density function (PDF) is:

mean = ab + θ

variance = ab²

Notation

The discrete geometric distribution applies to a sequence of independent Bernoulli experiments with an event of interest that has probability p.

Formula

If the random variable X is the total number of trials necessary to produce one event with probability p, then the probability mass function (PMF) of X is given by:

and X exhibits the following properties:

If the random variable Y is the number of nonevents that occur before the first event (with probability p) is observed, then the probability mass function (PMF) of Y is given by:

and Y exhibits the following properties:

Notation

The hypergeometric distribution is used for samples drawn from small populations, without replacement. For example, you have a shipment of N televisions, where N₁ are good (successes) and N₂ are defective (failure). If you sample n televisions of N at random, without replacement, you can find the probability that exactly x of the n televisions are good.

Formula

The probability mass function (PMF) is:

Notation

The integer distribution is a discrete uniform distribution on a set of integers. Each integer has equal probability of occurring.

The normal distribution (also called Gaussian distribution) is the most used statistical distribution because of the many physical, biological, and social processes that it can model.

Formula

The probability density function (PDF) is:

The cumulative distribution function (CDF) is:

mean = μ

variance = σ ²

standard deviation = σ

Notation

The Laplace distribution is used when the distribution is more peaked than a normal distribution.

Formula

The probability density function (PDF) is:

mean = a

variance = 2b²

Notation

Use the largest extreme value distribution to model the largest value from a distribution. If you have a sequence of exponential distributions, and X_(n) is the maximum of the first n, then X_(n) – ln(n) converges in distribution to the largest extreme value distribution. Thus, for large values of n, the largest extreme value distribution is a good approximation to the distribution of X_(n) – ln(n).

Formula

The probability density function (PDF) is:

The cumulative distribution function (CDF) is:

mean = μ + γσ

variance = π ² σ ² / 6

Notation

A continuous distribution that is symmetric, similar to the normal distribution, but with heavier tails.

Formula

The probability density function (PDF) is:

The cumulative distribution function (CDF) is:

mean = μ

Notation

A variable x has a loglogistic distribution with threshold λ if Y = log (x – λ) has a logistic distribution.

Formula

The probability density function (PDF) is:

The cumulative distribution function (CDF) is:

when σ < 1:

when σ < 1/2:

Notation

A variable x has a lognormal distribution if log(x – λ ) has a normal distribution.

Formula

The probability density function (PDF) is:

The cumulative distribution function (CDF) is:

Notation

The discrete negative binomial distribution applies to a series of independent Bernoulli experiments with an event of interest that has probability p.

Formula

If the random variable X is the number of trials necessary to produce r events that each have probability p, then the probability mass function (PMF) of X is given by:

and X exhibits the following properties:

If the random variable Y is the number of nonevents that occur before you observe the r events, which each have probability p, then the probability mass function (PMF) of Y is given by:

and Y exhibits the following properties:

Notation

The Poisson distribution is a discrete distribution that models the number of events based on a constant rate of occurrence. The Poisson distribution can be used as an approximation to the binomial when the number of independent trials is large and the probability of success is small.

Formula

The probability mass function (PMF) is:

mean = λ

variance = λ

Notation

Use the smallest extreme value distribution to model the smallest value from a distribution. If Y follows the Weibull distribution, then log(Y) follows the smallest extreme value distribution.

Formula

The probability density function (PDF) is:

The cumulative distribution function (CDF) is:

Notation

Formula

The probability density function (PDF) is:

mean = 0, when ν > 0

Notation

The PDF of the triangular distribution has a triangular shape.

Formula

The probability density function (PDF) is:

Notation

The uniform distribution characterizes data over an interval uniformly, with a as the smallest value and b as the largest value.

Formula

The probability density function (PDF) is:

Notation

The Weibull distribution is useful to model product failure times.

Formula

The probability density function (PDF) is:

The cumulative distribution function (CDF) is:

Notation