Chi-Squared (χ²) Distribution

The distribution of the sum of squared standard normal deviates.

The "Goodness of Fit" Distribution

The Chi-Squared (χ²) distribution is a continuous probability distribution that is widely used in hypothesis testing. It arises as the distribution of a sum of squared independent standard normal random variables.

In finance and econometrics, it is the backbone of the Chi-Squared test, which is used to test the goodness of fit of a model, check for independence between categorical variables, and compare variances. For instance, a risk manager might use it to test if the observed frequency of portfolio losses matches the frequency predicted by their risk model.

Interactive χ² Distribution

Adjust the degrees of freedom to see how the shape of the distribution changes. Notice how it becomes more symmetric and bell-shaped as the degrees of freedom increase.

Degrees of Freedom (k): 5

Mean (

\mu

): 5.00

Variance (

\sigma^2

): 10.00

Core Concepts

Probability Density Function (PDF)

The PDF is defined by one parameter: the degrees of freedom (

k

f(x; k) = \frac{1}{2^{k/2}\Gamma(k/2)} x^{k/2-1} e^{-x/2}

$x \ge 0$ is the variable.
$k$ represents the degrees of freedom.
$\Gamma(k/2)$ is the Gamma function.

Expected Value & Variance

Expected Value (Mean)

E[X] = k

Variance

Var(X) = 2k

Key Derivations

Deriving the Moments via the Gamma Distribution

The moments of the Chi-Squared distribution are most easily derived by recognizing it as a special case of the Gamma distribution.

Step 1: Connect Chi-Squared to Gamma

A Chi-Squared distribution with $k$ degrees of freedom, $\chi^2(k)$ , is equivalent to a Gamma distribution with shape parameter $\alpha = k/2$ and rate parameter $\beta = 1/2$ .

We know from the Gamma Distribution page that for a Gamma distribution $\text{Gamma}(\alpha, \beta)$ , the moments are:

$E[X] = \frac{\alpha}{\beta}$
$Var(X) = \frac{\alpha}{\beta^2}$

Deriving the Expected Value (Mean)

Step 2: Substitute Gamma Parameters

Substitute $\alpha = k/2$ and $\beta = 1/2$ into the mean formula for the Gamma distribution.

E[X] = \frac{k/2}{1/2}

Step 3: Simplify

The 1/2 terms cancel out, leaving us with a remarkably simple result.

Final Mean Formula

E[X] = k

Deriving the Variance

Step 4: Substitute Gamma Parameters

Substitute $\alpha = k/2$ and $\beta = 1/2$ into the variance formula for the Gamma distribution.

Var(X) = \frac{k/2}{(1/2)^2} = \frac{k/2}{1/4}

Step 5: Simplify

Multiplying by the reciprocal of 1/4 gives the final result.

Final Variance Formula

Var(X) = 2k