ThmDex – An index of mathematical definitions, results, and conjectures.

Result R5435 on D212: Chi-squared random unsigned real number

Expectation of a chi-squared random unsigned real number

Formulation 0

Let $\chi \in \text{ChiSquared}(N)$ be a D212: Chi-squared random unsigned real number.

Then \begin{equation} \mathbb{E} \chi = N \end{equation}

Proofs

Proof 0

Let $\chi \in \text{ChiSquared}(N)$ be a D212: Chi-squared random unsigned real number.

Let $Z_1, \ldots, Z_N \in \text{Gaussian}(0, 1)$ be independent standard gaussians. Using R5444: Expectation of a squared standard gaussian random real number, we have \begin{equation} \mathbb{E} \chi = \mathbb{E} \left( \sum_{n = 1}^N Z^2_n \right) = \sum_{n = 1}^N \mathbb{E} Z^2_n = \sum_{n = 1}^N 1 = N \end{equation} $\square$