where Γ(k/2) denotes the Gamma function, which has closed-form values at the half-integers.
It is very interesting as the sum of squares of n independent standard random variables has the chi-square distribution of n-1 freedom.
3. Student's t distribution
Student's t-distribution is the probability distribution of the ratio
where
- Z is normally distributed with expected value 0 and variance 1;
- V has a chi-square distribution with ν degrees of freedom;
- Z and V are independent.
Student's t-distribution has the probability density function
where ν is the number of degrees of freedom and Γ is the Gamma function.
Asymptotically , if ν goes to infinity (sample length increases), the distribution gives to a normal distribution.
4. F distribution
A random variate of the F-distribution arises as the ratio of two chi-squared variates:
where
- U1 and U2 have chi-square distributions with d1 and d2 degrees of freedom respectively, and
- U1 and U2 are independent (see Cochran's theorem for an application).
The probability density function of an F(d1, d2) distributed random variable is given by
for real x ≥ 0, where d1 and d2 are positive integers, and B is the beta function.
The cumulative distribution function is
where I is the regularized incomplete beta function.
Fisher-Snedecor Probability density function Cumulative distribution function parameters: 0,\ d_2>0" src="http://upload.wikimedia.org/math/6/c/3/6c3df1080d0ee2d5084dd8485a1e7ca8.png" style="border-top-style: none; border-right-style: none; border-bottom-style: none; border-left-style: none; border-width: initial; border-color: initial; vertical-align: middle; "> deg. of freedom
support: pdf: cdf: mean: for d2 > 2
median: mode: for d1 > 2
variance: for d2 > 4
skewness:
for d2 > 6
