The random variate \(X\) defined for the range \(0 \leq X \leq +\infty\), is said to have a Chi Distribution (i.e. \(X \sim \chi \left( n, \sigma \right)\)) with shape parameter \(n \in \mathbb{N}^+\) and scale parameter \(\sigma \in \mathbb{R}_0^+\).
This chapter uses a scaled parameterization. If \(Z \sim \chi^2(n)\), then \(X=\sigma\sqrt{Z/n}\) follows \(\chi(n,\sigma)\) in this notation.
\[ \text{f}(X) = \frac{2 \left( \frac{n}{2} \right)^{\frac{n}{2}} X^{n-1} e^{-\left( \frac{n}{2 \sigma^2} \right) X^2 } }{ \mathrm{ \Gamma} \left[ \frac{n}{2} \right] \sigma^n } \]
\[ \mu_j' = \left( \frac{2}{n} \right)^{\frac{j}{2}} \sigma^j \frac{\mathrm{\Gamma \left[ \frac{n+j}{2} \right] }}{\mathrm{\Gamma \left[ \frac{n}{2} \right]}} \]
\[ \text{Mo}(X) = \sigma \sqrt{\frac{n-1}{n}} \]
\[ \text{E}(X) = \sqrt{\frac{2}{n}} \, \sigma \, \frac{\Gamma\left(\frac{n+1}{2}\right)}{\Gamma\left(\frac{n}{2}\right)} \]
\[ \text{V}(X) = \sigma^2 \left[1 - \frac{2}{n}\left(\frac{\Gamma\left(\frac{n+1}{2}\right)}{\Gamma\left(\frac{n}{2}\right)}\right)^2\right] \]
There is no simple closed form. A useful expression is
\[ \text{Med}(X) = \sigma \sqrt{\frac{q_{\chi^2(n)}(0.5)}{n}} \]
where \(q_{\chi^2(n)}(0.5)\) is the 50th percentile of the standard Chi-squared distribution with \(n\) degrees of freedom.
n <- 6
sigma <- 1.2
x <- seq(0, 3, length.out = 1000)
fx <- dchisq(n * x^2 / sigma^2, df = n) * (2 * n * x / sigma^2)
plot(x, fx, type = "l", lwd = 2, col = "steelblue",
xlab = "x", ylab = "f(x)",
main = "Chi distribution",
sub = "(n = 6, sigma = 1.2)")
If the following is true
\[ \begin{align*} \begin{cases} \chi^2(n, \sigma) \text{ denotes a Chi-squared Distribution} \\ \text{U}(0,1) \text{ denotes a Uniform Distribution} \\ \text{N}(0,1) \text{ denotes a Standard Normal Distribution} \\ \text{N} \left( 0, \sigma^2 \right) \text{ denotes a Normal Distribution with $\mu = 0$ and variance $\sigma^2$} \end{cases} \end{align*} \]
and \(\chi^2(n,\sigma) \sim -2 \sigma^2 \text{ln} \left( \prod_{i=1}^{r} \text{U}_i(0,1) \right)\) with \(r=\frac{n}{2}\) and \(n\) is even
and \(\chi^2(n,\sigma) \sim -2 \sigma^2 \text{ln} \left( \prod_{i=1}^{r} \text{U}_i(0,1) \right) + \left( \text{N}\left( 0, \sigma^2 \right) \right)^2\) with \(r=\frac{n-1}{2}\) and \(n\) is odd
then it follows that
\[ \chi(n, \sigma) \sim \sqrt{\frac{\chi^2(n,\sigma)}{n}} \]
For \(n \rightarrow +\infty\) the Chi Distribution \(\chi(n,\sigma)\) is symmetric around \(\sigma\).
For \(X \sim \chi(n=6,\sigma=1.2)\), the probability that \(X \le 1.5\) is:
n <- 6
sigma <- 1.2
x0 <- 1.5
p <- pchisq(n * x0^2 / sigma^2, df = n)
cat("P(X <= 1.5) =", p, "\n")P(X <= 1.5) = 0.8464394 The Chi distribution appears in norms and root-mean-square quantities of normal vectors. It is frequently used in signal magnitude modeling, geometric length problems, and as a transformation companion to the Chi-squared distribution.