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 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.
The beta distribution is often used to represent processes with natural lower and upper limits.
The probability density function (PDF) is:
Term | Description |
---|---|
α | shape parameter 1 |
β | shape parameter 2 |
Γ | gamma function |
a | lower limit |
b | upper limit |
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.
mean = np
variance = np(1 – p)
The probability mass function (PMF) is:
Where equals .
In general, you can calculate k! as
Term | Description |
---|---|
n | number of trials |
x | number of events |
p | event probability |
The Cauchy distribution is symmetric around zero, but the tails approach zero less quickly than do those of the normal distribution.
The probability density function (PDF) is:
Term | Description |
---|---|
a | location parameter |
b | scale parameter |
π | Pi (~3.142) |
If you do not specify values, Minitab uses a = 0 and b = 1.
If X has a standard normal distribution, X^{2} has a chi-square distribution with one degree of freedom, allowing it to be a commonly used sampling distribution.
The sum of n independent X^{2} 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.
The probability density function (PDF) is:
mean = v
variance = 2v
Term | Description |
---|---|
ν | degrees of freedom |
Γ | gamma function |
e | base of the natural logarithm |
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 |
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.
The probability density function (PDF) is:
The cumulative distribution function (CDF) is:
mean = θ + λ
variance = θ^{2}
Term | Description |
---|---|
θ | scale parameter |
λ | threshold parameter |
exp | base of the natural logarithm |
Some references use 1 / θ for a parameter.
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.
The probability density function (PDF) is:
Term | Description |
---|---|
Γ | gamma function |
u | numerator degrees of freedom |
v | denominator degrees of freedom |
The gamma distribution is often used to model positively skewed data.
The probability density function (PDF) is:
mean = ab + θ
variance = ab^{2}
Term | Description |
---|---|
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 |
Some references use 1/ b for a parameter.
The discrete geometric distribution applies to a sequence of independent Bernoulli experiments with an event of interest that has probability p.
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:
Term | Description |
---|---|
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 |
The hypergeometric distribution is used for samples drawn from small populations, without replacement. For example, you have a shipment of N televisions, where N_{1} are good (successes) and N_{2 }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.
The probability mass function (PMF) is:
Term | Description |
---|---|
N | N_{1} + N_{2} = population size |
N_{1} | number of events in the population |
N_{2} | number of non-events in the population |
n | sample size |
x | number of events in the sample |
The integer distribution is a discrete uniform distribution on a set of integers. Each integer has equal probability of occurring.
The Laplace distribution is used when the distribution is more peaked than a normal distribution.
The probability density function (PDF) is:
mean = a
variance = 2b^{2}
Term | Description |
---|---|
a | location parameter |
b | scale parameter |
e | base of natural logarithm |
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).
The probability density function (PDF) is:
The cumulative distribution function (CDF) is:
mean = μ + γσ
variance = π ^{2} σ ^{2} / 6
Term | Description |
---|---|
σ | scale parameter |
μ | location parameter |
γ | Euler constant (~0.57722) |
A continuous distribution that is symmetric, similar to the normal distribution, but with heavier tails.
The probability density function (PDF) is:
The cumulative distribution function (CDF) is:
mean = μ
Term | Description |
---|---|
μ | location parameter |
σ | scale parameter |
A variable x has a loglogistic distribution with threshold λ if Y = log (x – λ) has a logistic distribution.
The probability density function (PDF) is:
The cumulative distribution function (CDF) is:
when σ < 1:
when σ < 1/2:
Term | Description |
---|---|
μ | location parameter |
σ | scale parameter |
λ | threshold parameter |
Γ | gamma function |
exp | base of the natural logarithm |
A variable x has a lognormal distribution if log(x – λ ) has a normal distribution.
The probability density function (PDF) is:
The cumulative distribution function (CDF) is:
Term | Description |
---|---|
μ | location parameter |
σ | scale parameter |
λ | threshold parameter |
π | Pi (~3.142) |
The discrete negative binomial distribution applies to a series of independent Bernoulli experiments with an event of interest that has probability p.
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:
This negative binomial distribution is also known as the Pascal distribution.
Term | Description |
---|---|
X | Y + r |
r | number of events |
p | probability of an event |
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.
The probability density function (PDF) is:
The cumulative distribution function (CDF) is:
mean = μ
variance = σ ^{2}
standard deviation = σ
Term | Description |
---|---|
exp | base of the natural logarithm |
π | Pi (~3.142) |
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.
The probability mass function (PMF) is:
mean = λ
variance = λ
Term | Description |
---|---|
e | base of the natural logarithm |
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.
The probability density function (PDF) is:
The cumulative distribution function (CDF) is:
Term | Description |
---|---|
ξ | location parameter |
θ | scale parameter |
e | base of the natural logarithm |
v | Euler constant (~0.57722) |
mean = 0, when ν > 0
Term | Description |
---|---|
Γ | gamma function |
v | degrees of freedom |
π | Pi (~3.142) |
The PDF of the triangular distribution has a triangular shape.
The probability density function (PDF) is:
Term | Description |
---|---|
a | lower endpoint |
b | upper endpoint |
c | mode (location where the PDF peaks) |
The uniform distribution characterizes data over an interval uniformly, with a as the smallest value and b as the largest value.
The probability density function (PDF) is:
Term | Description |
---|---|
a | lower endpoint |
b | upper endpoint |
The Weibull distribution is useful to model product failure times.
The probability density function (PDF) is:
The cumulative distribution function (CDF) is:
Term | Description |
---|---|
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 |
Some references use 1/α as a parameter.