Probability density function for a chi-squared random ...

Result on D212: Chi-squared random unsigned real number

Probability density function for a chi-squared random unsigned real number

Formulation 0

Let $X \in \text{ChiSquared}(n)$ be a D212: Chi-squared random unsigned real number.
Let $B \in \mathcal{B}(\mathbb{R})$ be a D5113: Standard real Borel set.

Then \begin{equation} \mathbb{P}(X \in B) = \int_B \frac{1}{2^{n / 2} \Gamma( n / 2 )} t^{\frac{n}{2} - 1} e^{- t / 2} I_{[0, \infty)} (t) \, d t \end{equation}

Formulation 1

Let $X \in \text{ChiSquared}(n)$ be a D212: Chi-squared random unsigned real number such that

(i)	$\mu_X$ is a D204: Probability distribution measure for $X$

Let $\ell$ be the D5645: Real Lebesgue measure.
Let $t \in \mathbb{R}$ be a D993: Real number.

Then \begin{equation} \frac{d \mu_X}{d \ell} (t) = \frac{1}{2^{n / 2} \Gamma( n / 2 )} t^{\frac{n}{2} - 1} e^{- t / 2} I_{[0, \infty)} (t) \end{equation}

Proofs