Chi Distribution — Definition, Formula & Examples

The chi distribution is the probability distribution of the square root of a chi-squared random variable. It describes the distribution of the Euclidean distance from the origin of a point drawn from a standard multivariate normal distribution.

If $Q \sim \chi^2_k$ , then $X = \sqrt{Q}$ follows a chi distribution with $k$ degrees of freedom, denoted $X \sim \chi_k$ . Its support is $x \in [0, \infty)$ , and it belongs to the family of continuous probability distributions.

Key Formula

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

Where:

$x$ = Value of the chi-distributed random variable (x ≥ 0)
$k$ = Degrees of freedom (positive integer)
$\Gamma$ = Gamma function, a generalization of the factorial

How It Works

The chi distribution arises when you take the square root of a sum of squared independent standard normal variables. If

Z_1, Z_2, \ldots, Z_k

are independent standard normal random variables, then

X = \sqrt{Z_1^2 + Z_2^2 + \cdots + Z_k^2}

follows a chi distribution with

k

degrees of freedom. The mean and variance depend on

k

through the gamma function. For large

k

, the distribution approaches a normal shape centered near

\sqrt{k}

Worked Example

Problem: Let X follow a chi distribution with k = 2 degrees of freedom. Compute the PDF at x = 1.

Write the PDF: Substitute k = 2 into the chi PDF formula.

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

Simplify constants: We have 2^0 = 1, 1^1 = 1, and Γ(1) = 0! = 1.

f(1; 2) = \frac{1 \cdot 1 \cdot e^{-0.5}}{1} = e^{-0.5}

Evaluate: Compute the numerical value.

f(1; 2) \approx 0.6065

Answer: The PDF of the chi distribution with 2 degrees of freedom evaluated at x = 1 is approximately 0.6065.

Why It Matters

The chi distribution appears in directional statistics and when analyzing the magnitude of multivariate normal vectors. It is used in radar signal processing, error analysis in GPS positioning, and any setting where you care about the distance of a random point from the origin rather than the squared distance.

Common Mistakes

Mistake: Confusing the chi distribution with the chi-squared distribution.

Correction: The chi distribution is the square root of a chi-squared variable. Their PDFs, means, and variances are different. If Q ~ χ²_k, then √Q ~ χ_k — they are related but distinct distributions.