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Probability density function of the chi-squared distribution | The Book of Statistical Proofs

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The Book of Statistical Proofs – a centralized, open and collaboratively edited archive of statistical theorems for the computational sciences

Index: The Book of Statistical Proofs ▷ Probability Distributions ▷ Univariate continuous distributions ▷ Chi-squared distribution ▷ Probability density function Theorem: Let $Y$ be a random variable following a chi-squared distribution : \[\label{eq:chi2} Y \sim \chi^{2}(k) \; .\] Then, the probability density function of $Y$ is \[\label{eq:chi2-pdf} f_Y(y) = \frac{1}{2^{k/2} \, \Gamma (k/2)} \, y^{k/2-1} \exp \left[ -y/2 \right] \; .\] Proof: A chi-square-distributed random variable with $k$ degrees of freedom is defined as the sum of $k$ squared standard normal r

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