Chi-squared Distribution - Relation To Other Distributions

Relation To Other Distributions

• As, (normal distribution)

• (Noncentral chi-squared distribution with non-centrality parameter )

• If then has the chi-squared distribution

• As a special case, if then has the chi-squared distribution

• (The squared norm of k standard normally distributed variables is a chi-squared distribution with k degrees of freedom)

• If and, then . (gamma distribution)

• If then (chi distribution)

• If, then is an exponential distribution. (See Gamma distribution for more.)

• If (Rayleigh distribution) then

• If (Maxwell distribution) then

• If then (Inverse-chi-squared distribution)

• The chi-squared distribution is a special case of type 3 Pearson distribution

• If and are independent then (beta distribution)

• If (uniform distribution) then

• is a transformation of Laplace distribution

• If then

• chi-squared distribution is a transformation of Pareto distribution

• Student's t-distribution is a transformation of chi-squared distribution

• Student's t-distribution can be obtained from chi-squared distribution and normal distribution

• Noncentral beta distribution can be obtained as a transformation of chi-squared distribution and Noncentral chi-squared distribution

• Noncentral t-distribution can be obtained from normal distribution and chi-squared distribution

A chi-squared variable with k degrees of freedom is defined as the sum of the squares of k independent standard normal random variables.

If Y is a k-dimensional Gaussian random vector with mean vector μ and rank k covariance matrix C, then X = (Y−μ)TC−1(Y−μ) is chi-squared distributed with k degrees of freedom.

The sum of squares of statistically independent unit-variance Gaussian variables which do not have mean zero yields a generalization of the chi-squared distribution called the noncentral chi-squared distribution.

If Y is a vector of k i.i.d. standard normal random variables and A is a k×k idempotent matrix with rank k−n then the quadratic form YTAY is chi-squared distributed with k−n degrees of freedom.

The chi-squared distribution is also naturally related to other distributions arising from the Gaussian. In particular,

• Y is F-distributed, Y ~ F(k₁,k₂) if where X₁ ~ χ²(k₁) and X₂ ~ χ²(k₂) are statistically independent.

• If X is chi-squared distributed, then is chi distributed.

• If X₁ ~ χ2_k1 and X₂ ~ χ2_k2 are statistically independent, then X₁ + X₂ ~ χ2_k1+k2. If X₁ and X₂ are not independent, then X₁ + X₂ is not chi-squared distributed.