Neha Patil (Editor)

Chi distribution

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Chi distribution

Parameters
  
k > 0 {displaystyle k>0,} (degrees of freedom)

Support
  
x ∈ [ 0 ; ∞ ) {displaystyle xin [0;infty )}

PDF
  
2 1 − k / 2 x k − 1 e − x 2 / 2 Γ ( k / 2 ) {displaystyle { rac {2^{1-k/2}x^{k-1}e^{-x^{2}/2}}{Gamma (k/2)}}}

CDF
  
P ( k / 2 , x 2 / 2 ) {displaystyle P(k/2,x^{2}/2),}

Mean
  
μ = 2 Γ ( ( k + 1 ) / 2 ) Γ ( k / 2 ) {displaystyle mu ={sqrt {2}},{ rac {Gamma ((k+1)/2)}{Gamma (k/2)}}}

Mode
  
k − 1 {displaystyle {sqrt {k-1}},} for k ≥ 1 {displaystyle kgeq 1}

In probability theory and statistics, the chi distribution is a continuous probability distribution. It is the distribution of the square root of the sum of squares of independent random variables having a standard normal distribution, or equivalently, the distribution of the Euclidean distance of the random variables from the origin. The most familiar examples are the Rayleigh distribution with chi distribution with 2 degrees of freedom, and the Maxwell distribution of (normalized) molecular speeds which is a chi distribution with 3 degrees of freedom (one for each spatial coordinate). If X i are k independent, normally distributed random variables with means μ i and standard deviations σ i , then the statistic

Contents

Y = ∑ i = 1 k ( X i − μ i σ i ) 2

is distributed according to the chi distribution. Accordingly, dividing by the mean of the chi distribution (scaled by the square root of n − 1) yields the correction factor in the unbiased estimation of the standard deviation of the normal distribution. The chi distribution has one parameter: k which specifies the number of degrees of freedom (i.e. the number of X i ).

Probability density function

The probability density function is

f ( x ; k ) = 2 1 − k 2 x k − 1 e − x 2 2 Γ ( k 2 )

where Γ ( z ) is the Gamma function.

Cumulative distribution function

The cumulative distribution function is given by:

F ( x ; k ) = P ( k / 2 , x 2 / 2 )

where P ( k , x ) is the regularized Gamma function.

Moment generating function

The moment generating function is given by:

M ( t ) = M ( k 2 , 1 2 , t 2 2 ) + t 2 Γ ( ( k + 1 ) / 2 ) Γ ( k / 2 ) M ( k + 1 2 , 3 2 , t 2 2 )

where M ( a , b , z ) is Kummer's confluent hypergeometric function.

Characteristic function

The characteristic function is given by:

φ ( t ; k ) = M ( k 2 , 1 2 , − t 2 2 ) + i t 2 Γ ( ( k + 1 ) / 2 ) Γ ( k / 2 ) M ( k + 1 2 , 3 2 , − t 2 2 )

where again, M ( a , b , z ) is Kummer's confluent hypergeometric function.

Properties

Differential equation

{ x f ′ ( x ) + f ( x ) ( − ν + x 2 + 1 ) = 0 , f ( 1 ) = 2 1 − ν 2 e Γ ( ν 2 ) }

Moments

The raw moments are then given by:

μ j = 2 j / 2 Γ ( ( k + j ) / 2 ) Γ ( k / 2 )

where Γ ( z ) is the Gamma function. The first few raw moments are:

μ 1 = 2 Γ ( ( k + 1 ) / 2 ) Γ ( k / 2 ) μ 2 = k μ 3 = 2 2 Γ ( ( k + 3 ) / 2 ) Γ ( k / 2 ) = ( k + 1 ) μ 1 μ 4 = ( k ) ( k + 2 ) μ 5 = 4 2 Γ ( ( k + 5 ) / 2 ) Γ ( k / 2 ) = ( k + 1 ) ( k + 3 ) μ 1 μ 6 = ( k ) ( k + 2 ) ( k + 4 )

where the rightmost expressions are derived using the recurrence relationship for the Gamma function:

Γ ( x + 1 ) = x Γ ( x )

From these expressions we may derive the following relationships:

Mean: μ = 2 Γ ( ( k + 1 ) / 2 ) Γ ( k / 2 )

Variance: σ 2 = k − μ 2

Skewness: γ 1 = μ σ 3 ( 1 − 2 σ 2 )

Kurtosis excess: γ 2 = 2 σ 2 ( 1 − μ σ γ 1 − σ 2 )

Entropy

The entropy is given by:

S = ln ⁡ ( Γ ( k / 2 ) ) + 1 2 ( k − ln ⁡ ( 2 ) − ( k − 1 ) ψ 0 ( k / 2 ) )

where ψ 0 ( z ) is the polygamma function.

  • If X ∼ χ k ( x ) then X 2 ∼ χ k 2 (chi-squared distribution)
  • lim k → ∞ χ k ( x ) − μ k σ k → d   N ( 0 , 1 ) (Normal distribution)
  • If X ∼ N ( 0 , 1 ) then | X | ∼ χ 1 ( x )
  • If X ∼ χ 1 ( x ) then σ X ∼ H N ( σ ) (half-normal distribution) for any σ > 0
  • χ 2 ( x ) ∼ R a y l e i g h ( 1 ) (Rayleigh distribution)
  • χ 3 ( x ) ∼ M a x w e l l ( 1 ) (Maxwell distribution)
  • ∥ N i = 1 , … , k ( 0 , 1 ) ∥ 2 ∼ χ k ( x ) (The 2-norm of k standard normally distributed variables is a chi distribution with k degrees of freedom)
  • chi distribution is a special case of the generalized gamma distribution or the Nakagami distribution or the noncentral chi distribution

    • References

    Chi distribution Wikipedia
    (Text) CC BY-SA


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