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  "body": "\n## Definition\n\nLet $k$ be a positive integer and let $\\mu_1, \\dots, \\mu_k$ be real numbers.\nLet $X_1, \\dots, X_k$ be [[independent|0E9502NT92]] random variables with $X_i \\sim \\mathcal{N}(\\mu_i, 1)$, the [[normal distribution|07RXBVTVZB]] with mean $\\mu_i$ and variance $1$.\nThe distribution of\n$$Q = \\sum_{i=1}^k X_i^2$$\ndepends only on $k$ and on $\\lambda = \\sum_{i=1}^k \\mu_i^2$.\nIt is the *noncentral chi-squared distribution* with $k$ degrees of freedom and noncentrality parameter $\\lambda$, written $\\chi'^2_k(\\lambda)$.\n\n## Remarks\n\nFor $\\lambda = 0$ it is the [[chi-squared distribution|0RFMMBHFTV]] with $k$ degrees of freedom.\nIts mean is $k + \\lambda$ and its variance is $2(k + 2\\lambda)$.\nIts density is a [[Poisson|0VEFSSKX18]] mixture of central chi-squared densities:\n$$f(x) = \\sum_{j=0}^{\\infty} e^{-\\lambda/2} \\frac{(\\lambda/2)^j}{j!} f_{k + 2j}(x), \\quad x \u003e 0,$$\nwhere $f_m$ is the chi-squared density with $m$ degrees of freedom.\n\nExample: 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 $k + \\lambda$.\n\nNon-example: the variances must be $1$.\nIf $X \\sim \\mathcal{N}(0, 4)$, then $X^2$ has mean $4$ and variance $32$.\nA $\\chi'^2_1(\\lambda)$ 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.\n",
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