Statistics - Chi-squared Distribution

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The chi-squared distribution (chi-square or ${X^2}$ - distribution) with degrees of freedom, k is the distribution of a sum of the squares of k independent standard normal random variables. It is one of the most widely used probability distributions in statistics. It is a special case of the gamma distribution.

Chi-squared distribution is widely used by statisticians to compute the following:

Estimation of Confidence interval for a population standard deviation of a normal distribution using a sample standard deviation.
To check independence of two criteria of classification of multiple qualitative variables.
To check the relationships between categorical variables.
To study the sample variance where the underlying distribution is normal.
To test deviations of differences between expected and observed frequencies.
To conduct a The chi-square test (a goodness of fit test).

Probability density function

Probability density function of Chi-Square distribution is given as:

Formula

${ f(x; k ) = }$

$\begin {cases} \frac{x^{ \frac{k}{2} - 1} e^{-\frac{x}{2}}}{2^{\frac{k}{2}}\Gamma(\frac{k}{2})}, & \text{if $x \gt 0 $} \\[7pt] 0, & \text{if $x \le 0 $} \end{cases}$

Where −

${\Gamma(\frac{k}{2})}$ = Gamma function having closed form values for integer parameter k.
${x}$ = random variable.
${k}$ = integer parameter.

Cumulative distribution function

Cumulative distribution function of Chi-Square distribution is given as:

Formula

${ F(x; k) = \frac{\gamma(\frac{x}{2}, \frac{k}{2})}{\Gamma(\frac{k}{2})}\\[7pt] = P (\frac{x}{2}, \frac{k}{2}) }$

Where −

${\gamma(s,t)}$ = lower incomplete gamma function.
${P(s,t)}$ = regularized gamma function.
${x}$ = random variable.
${k}$ = integer parameter.

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