diff --git a/P/norm-lincomb.md b/P/norm-lincomb.md index b1170ce..11f2489 100644 --- a/P/norm-lincomb.md +++ b/P/norm-lincomb.md @@ -21,7 +21,7 @@ username: "JoramSoch" --- -**Theorem:** Let $X_1, \ldots, X_n$ be [independent](/D/ind) [normally distributed](/D/norm) [random variables](/D/rvar) with [means](/D/mean) $\mu_1, \ldots, \mu_n$ and [variances](/D/var) $\sigma^2_1, \ldots, \sigma^2_n$: +**Theorem:** Let $X_1, \ldots, X_n$ be [uncorrelated](D/cov) [normally distributed](/D/norm) [random variables](/D/rvar) with [means](/D/mean) $\mu_1, \ldots, \mu_n$ and [variances](/D/var) $\sigma^2_1, \ldots, \sigma^2_n$: $$ \label{eq:norm} X_i \sim \mathcal{N}(\mu_i, \sigma^2_i) \quad \text{for} \quad i = 1, \ldots, n \; . @@ -79,4 +79,4 @@ $$ \label{eq:A-b-mu-Sigma} A \mu &= \left[ a_1, \ldots, a_n \right] \left[ \begin{array}{c} \mu_1 \\ \vdots \\ \mu_n \end{array} \right] = \sum_{i=1}^{n} a_i \mu_i \quad \text{and} \quad \\ A \Sigma A^\mathrm{T} &= \left[ a_1, \ldots, a_n \right] \left[ \begin{array}{ccc} \sigma^2_1 & \cdots & 0 \\ \vdots & \ddots & \vdots \\ 0 & \cdots & \sigma^2_n \end{array} \right] \left[ \begin{array}{c} a_1 \\ \vdots \\ a_n \end{array} \right] = \sum_{i=1}^{n} a_i^2 \sigma^2_i \; . \end{split} -$$ \ No newline at end of file +$$