Methods and formulas for Probability Distributions

Select the method or formula of your choice.

Probability density function

The probability density function (PDF) of a random variable, X, allows you to calculate the probability of an event, as follows:
  • For continuous distributions, the probability that X has values in an interval (a, b) is precisely the area under its PDF in the interval (a, b).
  • For discrete distributions, the probability that X has values in an interval (a, b) is exactly the sum of the PDF (also called the probability mass function) of the possible discrete values of X in (a, b).
Use PDF to determine the value of the probability density function at a known value x of the random variable X.

Cumulative distribution function

The cumulative distribution function (CDF) calculates the cumulative probability for a given x-value. Use the CDF to determine the probability that a random observation that is taken from the population will be less than or equal to a certain value. You can also use this information to determine the probability that an observation will be greater than a certain value, or between two values.
  • For continuous distributions, the CDF gives the area under the probability density function, up to the x-value that you specify.
  • For discrete distributions, the CDF gives the cumulative probability for x-values that you specify.

Inverse cumulative probability

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 in the Session window (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.

Beta distribution

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

Formula

The probability density function (PDF) is:

Notation

TermDescription
α shape parameter 1
β shape parameter 2
Γgamma function
a lower limit
b upper limit

When a = 0, b = 1,

the PDF is:

Binomial distribution

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

TermDescription
n number of trials
x number of events
p event probability

Cauchy distribution

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

TermDescription
a location parameter
b scale parameter
π Pi (~3.142)
Note

If you do not specify values, Minitab uses a = 0 and b = 1.

Chi-square distribution

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

TermDescription
ν degrees of freedom
Γgamma function
e base of the natural logarithm

Discrete distribution

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.

Value Prob
−1 0.2
0 0.5
1 0.3

Exponential distribution

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

TermDescription
θ scale parameter
λ threshold parameter
exp base of the natural logarithm
Note

Some references use 1 / θ for a parameter.

F-distribution

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

TermDescription
Γgamma function
u numerator degrees of freedom
v denominator degrees of freedom

Gamma distribution

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

Formula

The probability density function (PDF) is:

mean = ab + θ

variance = ab2

Notation

TermDescription
a shape parameter (when a = 1, the gamma PDF is the same as the exponential PDF)
b scale parameter
θ threshold parameter
Γ gamma function
e base of the natural logarithm
Note

Some references use 1/ b for a parameter.

Geometric distribution

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

TermDescription
X number of trials to produce one event, Y + 1
Y number of nonevents that occur before the first event
p probability that an event occurs on each trial

Hypergeometric distribution

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

TermDescription
N N1 + N2 = population size
N1 number of events in the population
N2 number of non-events in the population
n sample size
x number of events in the sample

Integer distribution

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

Normal distribution

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

TermDescription
expbase of the natural logarithm
π Pi (~3.142)

Laplace distribution

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

TermDescription
a location parameter
b scale parameter
e base of natural logarithm

Largest extreme value distribution

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

TermDescription
σ scale parameter
μ location parameter
γ Euler constant (~0.57722)

Logistic distribution

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

TermDescription
μ location parameter
σ scale parameter

Loglogistic distribution

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

TermDescription
μ location parameter
σ scale parameter
λ threshold parameter
Γ gamma function
exp base of the natural logarithm

Lognormal distribution

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

Formula

The probability density function (PDF) is:

The cumulative distribution function (CDF) is:

Notation

TermDescription
μ location parameter
σ scale parameter
λ threshold parameter
π Pi (~3.142)

Negative binomial distribution

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:

Note

This negative binomial distribution is also known as the Pascal distribution.

Notation

TermDescription
X Y + r
r number of events
p probability of an event

Poisson distribution

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

TermDescription
e base of the natural logarithm

Smallest extreme value distribution

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

TermDescription
ξ location parameter
θ scale parameter
e base of the natural logarithm
v Euler constant (~0.57722)

t-distribution

The t-distribution converges to the normal distribution as the degrees of freedom increase. The t-distribution is useful to do the following:
  • Creating confidence intervals of the population mean from a normal distribution when the variance is unknown.
  • Determining whether two sample means from normal populations with unknown but equal variances are significantly different.
  • Testing the significance of regression coefficients.

Formula

The probability density function (PDF) is:

mean = 0, when ν > 0

Notation

TermDescription
Γgamma function
v degrees of freedom
π Pi (~3.142)

Triangular distribution

The PDF of the triangular distribution has a triangular shape.

Formula

The probability density function (PDF) is:

Notation

TermDescription
a lower endpoint
b upper endpoint
c mode (location where the PDF peaks)

Uniform distribution

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

TermDescription
a lower endpoint
b upper endpoint

Weibull distribution

The Weibull distribution is useful to model product failure times.

Formula

The probability density function (PDF) is:

The cumulative distribution function (CDF) is:

Notation

TermDescription
α scale parameter
β shape parameter, when β = 1 the Weibull PDF is the same as the exponential PDF
λ threshold parameter
Γ gamma function
exp base of the natural logarithm
Note

Some references use 1/α as a parameter.

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