with 𝑘degrees of freedom is the distribution of a sum of the squares of 𝑣independentstandard normal random variables
assume 𝑧 is a random variable with Standard Normal Distribution, then:
- 𝑧²: has distribution 𝐶𝘩𝑖-𝑆𝑞𝑢𝑎𝑟𝑒(𝑣=1)
- 𝑧₁² + 𝑧₂² + … + 𝑧_𝑘² : has distribution 𝐶𝘩𝑖-𝑆𝑞𝑢𝑎𝑟𝑒(𝑣=𝑘)

Probability Density Function

𝑓(𝑋=𝑥) = [1/(2^𝑣/2Γ(𝑣/2))] * 𝑥^𝑣/2-1 * 𝑒^-𝑥/2# for 0≤x≤∞

where:

relation to other distributions

Chi-square densities with 𝑣 = 1, 5, 10, and 30 degrees of freedom. Each distribution is right-skewed. For large 𝑣, it is approximately Normal

Expectation

𝐄(𝑋) = 𝑣

proof

𝐶𝘩𝑖-𝑆𝑞𝑢𝑎𝑟𝑒(𝑣) = 𝐺𝑎𝑚𝑚𝑎(𝛼=𝑣/2,𝜆=1/2)

𝐺𝑎𝑚𝑚𝑎(𝛼,𝜆) distribution has 𝐄(𝑋) = 𝛼/𝜆

therefore

𝐄(𝑋) = (𝑣/2) / (1/2)

𝐄(𝑋) = 𝑣

𝑉𝑎𝑟(𝑋) = 2𝑣

proof

𝐶𝘩𝑖-𝑆𝑞𝑢𝑎𝑟𝑒(𝑣) = 𝐺𝑎𝑚𝑚𝑎(𝛼=𝑣/2,𝜆=1/2)

𝐺𝑎𝑚𝑚𝑎(𝛼,𝜆) distribution has 𝑉𝑎𝑟(𝑋) = 𝛼/𝜆²

therefore

𝑉𝑎𝑟(𝑋) = (𝑣/2) / (1/2)²

𝑉𝑎𝑟(𝑋) = 2𝑣