Introduction

Recall that if $Z$ is a random variable having a standard normal distribution then $Z^2$ has a chi-square distribution with one degree of freedom. Furthermore, if $Z_1,Z_2, \dots ,Z_p$ are independent and each $Z_i$ is distributed $N(0,1)$ then the random variable $\sum^p_{i=1} Z^2_i$ has a chi-square distribution with $p$ degrees of freedom.

Suppose now that the means of the normal distributions are not zero. We wish to find the distributions of $Z^2_i$ and $\sum^p_{i=1} Z^2_i$.

Definition 6..1

Let $X_i$ be distributed as $N(\mu_i ,1), \ i=1,2,\dots ,p$. Then $X^2_i$ is said to have a non-central chi-square distribution with one degree of freedom and non-centrality parameter $\mu^2_i$, and $\sum^p_{i=1} Z^2_i$ has a non-central chi-square distribution with $p$ degrees of freedom and non-centrality parameter $\lambda$ where $\lambda = \sum ^p_{i=1} \mu^2_i$.

Notation

If $W$ has a non-central chi-square distribution, with $p$ degrees of freedom and non-centrality parameter $\lambda$ we will write $W \sim \chi^2_p(\lambda)$. Of course if $\lambda =0$ we have the usual $\chi^2$ distribution, sometimes called the central chi-square distribution.

The term non-central can also apply in the case of the t-distribution. Recall that $\frac{Z}{\sqrt{W/\nu}}$ where $Z \sim N(0,1)$, $W \sim \chi^2_{\nu}$ and $Z$ and $W$ are independent has a t-distribution with parameter $\nu$. Now when the variable in the numerator has a non-zero mean and the distribution is said to be non-central t.

[Non-central t and F distributions are defined in 5.4.]

A common use of the non-central distributions is in calculating the power of the $\chi^2$, $t$ and $F$ tests and in such applications as robustness studies.