Noncentral chi-squared distribution

Builds on Normal distribution Chi-squared distribution Independence of two random variables Poisson distribution

Definition

Let $k$ be a positive integer and let $\mu_1, \dots, \mu_k$ be real numbers. Let $X_1, \dots, X_k$ be independent random variables with $X_i \sim \mathcal{N}(\mu_i, 1)$ , the normal distribution with mean $\mu_i$ and variance $1$ . The distribution of $Q = \sum_{i=1}^k X_i^2$ depends only on $k$ and on $\lambda = \sum_{i=1}^k \mu_i^2$ . It is the noncentral chi-squared distribution with $k$ degrees of freedom and noncentrality parameter $\lambda$ , written $\chi'^2_k(\lambda)$ .

Remarks

For $\lambda = 0$ it is the chi-squared distribution with $k$ degrees of freedom. Its mean is $k + \lambda$ and its variance is $2(k + 2\lambda)$ . Its density is a Poisson mixture of central chi-squared densities: $f(x) = \sum_{j=0}^{\infty} e^{-\lambda/2} \frac{(\lambda/2)^j}{j!} f_{k + 2j}(x), \quad x > 0,$ where $f_m$ is the chi-squared density with $m$ degrees of freedom.

Example: if $X \sim \mathcal{N}(\mu, 1)$ , then $X^2 \sim \chi'^2_1(\mu^2)$ , and $E(X^2) = 1 + \mu^2$ agrees with the mean $k + \lambda$ .

Non-example: the variances must be $1$ . If $X \sim \mathcal{N}(0, 4)$ , then $X^2$ has mean $4$ and variance $32$ . A $\chi'^2_1(\lambda)$ variable with mean $4$ has $\lambda = 3$ and variance $2(1 + 6) = 14$ , so $X^2$ has no noncentral chi-squared distribution with one degree of freedom.

Reading path

To understand this page from the ground up, read these 81 pages in order. The foundations come first.

Construction of the reals from Cauchy sequences concept
Element definition
Factorial definition
Integers object
Ordered field definition
Proposition concept
Random experiment concept
Rational numbers object
Real numbers object
Set definition
Ring of integers statement
Disjunction definition
Implication definition
Predicate definition
Ordered pair definition
Union definition
Sample space definition
Existential quantifier definition
The rationals form a field statement
Universal quantifier definition
Cartesian product definition
Subset definition
The rationals form an ordered field statement
Uniqueness quantifier definition
Binary relation definition
Universal set definition
Order on the reals definition
Power set definition
Complement definition
Function definition
Interval definition
Indicator function definition
Injective function definition
Preimage of a set definition
Sigma-algebra definition
Countable set definition
Event definition
Borel sigma-algebra definition
Probability axioms concept
Probability measure definition
Dirac measure definition
Probability distribution definition
Probability space definition
Conditional probability definition
Random variable definition
Discrete random variable definition
Cumulative distribution function definition
Independence of two events definition
Law of a random variable definition
Random variable with a density definition
Probability mass function definition
Joint probability distribution definition
Independence of two random variables definition
Mutual independence of events definition
Expected value definition
Probability density function definition
Identically distributed random variables definition
Poisson distribution definition
Marginal distribution definition
Variance definition
Independent sigma-algebras definition
Linearity of expectation statement
Density of the normal distribution definition
Exponential distribution definition
Independence means the joint law is the product of the marginals statement
Independence of random variables definition
Covariance definition
Normal distribution definition
Gamma distribution definition
Expectation of a product of independent random variables statement
Variance of a sum statement
Independent and identically distributed random variables definition
Independent random variables are uncorrelated statement
Statistic definition
Variance of a sum of independent random variables statement
Estimator definition
Sample mean definition
Bias of an estimator definition
Sample variance definition
Chi-squared distribution definition
Noncentral chi-squared distribution definition

Tag	`E3TEYV0D89`
URI	`m2://E3TEYV0D89`
Kind	definition
Writing	W0 stub
MSC	62E15, 60E05
Defines	Noncentral chi-squared distribution
Uses	Normal distribution, Chi-squared distribution, Independence of two random variables, Poisson distribution

Noncentral chi-squared distribution

Definition

Remarks

Reading path

Further reading

What this page needs for W1

What links here