Index: The Book of Statistical ProofsProbability Distributions ▷ Univariate continuous distributions ▷ Normal distribution ▷ Linear combination

Theorem: Let $X_1, \ldots, X_n$ be independent normally distributed random variables with means $\mu_1, \ldots, \mu_n$ and variances $\sigma^2_1, \ldots, \sigma^2_n$:

\[\label{eq:norm} X_i \sim \mathcal{N}(\mu_i, \sigma^2_i) \quad \text{for} \quad i = 1, \ldots, n \; .\]

Then, any linear combination of those random variables

\[\label{eq:lincomb} Y = \sum_{i=1}^{n} a_i X_i \quad \text{where} \quad a_1, \ldots, a_n \in \mathbb{R}\]

also follows a normal distribution

\[\label{eq:norm-lincomb} Y \sim \mathcal{N}\left( \sum_{i=1}^{n} a_i \mu_i, \; \sum_{i=1}^{n} a_i^2 \sigma^2_i \right)\]

with mean and variance which are functions of the individual means and variances.

Proof: A set of $n$ independent normal random variables $X_1, \ldots, X_n$ is equivalent to an $n \times 1$ random vector $x$ following a multivariate normal distribution with a diagonal covariance matrix. Therefore, we can write

\[\label{eq:norm-mvn} X_i \sim \mathcal{N}(\mu_i, \sigma^2_i), \; i = 1, \ldots, n \quad \Rightarrow \quad x = \left[ \begin{array}{c} X_1 \\ \vdots \\ X_n \end{array} \right] \sim \mathcal{N}(\mu, \Sigma)\]

with mean vector and covariance matrix

\[\label{eq:mu-Sigma} \mu = \left[ \begin{array}{c} \mu_1 \\ \vdots \\ \mu_n \end{array} \right] \quad \text{and} \quad \Sigma = \left[ \begin{array}{ccc} \sigma^2_1 & \cdots & 0 \\ \vdots & \ddots & \vdots \\ 0 & \cdots & \sigma^2_n \end{array} \right] = \mathrm{diag}\left( \left[ \sigma^2_1, \ldots, \sigma^2_n \right] \right) \; .\]

Thus, we can apply the linear transformation theorem for the multivariate normal distribution

\[\label{eq:mvn-ltt} x \sim \mathcal{N}(\mu, \Sigma) \quad \Rightarrow \quad y = Ax + b \sim \mathcal{N}(A\mu + b, A \Sigma A^\mathrm{T})\]

with the constant matrix and vector

\[\label{eq:A-b} A = \left[ a_1, \ldots, a_n \right] \quad \text{and} \quad b = 0 \; .\]

This implies the following distribution the linear combination given by equation \eqref{eq:lincomb}:

\[\label{eq:norm-lincomb-p1} Y = Ax + b \sim \mathcal{N}(A\mu, A \Sigma A^\mathrm{T}) \; .\]

Finally, we note that

\[\label{eq:A-b-mu-Sigma} \begin{split} A \mu &= \left[ a_1, \ldots, a_n \right] \left[ \begin{array}{c} \mu_1 \\ \vdots \\ \mu_n \end{array} \right] = \sum_{i=1}^{n} a_i \mu_i \quad \text{and} \quad \\ A \Sigma A^\mathrm{T} &= \left[ a_1, \ldots, a_n \right] \left[ \begin{array}{ccc} \sigma^2_1 & \cdots & 0 \\ \vdots & \ddots & \vdots \\ 0 & \cdots & \sigma^2_n \end{array} \right] \left[ \begin{array}{c} a_1 \\ \vdots \\ a_n \end{array} \right] = \sum_{i=1}^{n} a_i^2 \sigma^2_i \; . \end{split}\]

Metadata: ID: P235 | shortcut: norm-lincomb | author: JoramSoch | date: 2021-06-02, 08:24.