I have the following problem:
The random vector $X$ is normally distributed with $X \sim \mathcal{N}(\mu,\Omega)$. $\mu $ is a column vector with $(\mu_1, \mu_2, \mu_3)$.
$\Omega$ is $3\times 3$ matrix. And $A$ is a $3\times 3$ matrix.
Find the distribution of a random vector $Y$ where $Y = AX$.
How can I tackle this? Can I do the affine transformation here?
$\endgroup$ 11 Answer
$\begingroup$$X \sim \mathcal{N}\left( \mu_X, \Omega \right) $.
\begin{align*} \mathbb{E}[Y] &= \mathbb{E}[AX]\\ &= A \mathbb{E}[X]\\ &= A \mu_X \\ \\ cov(Y) &= \mathbb{E} \left [ (Y - \mu_Y)(Y - \mu_Y)^T \right ]\\ &= \mathbb{E} \left [ (AX- A\mu_X)(AX - A\mu_X)^T \right ]\\ &= A \mathbb{E} \left [ (X- \mu_X)(X - \mu_X)^T \right ]A^T\\ &= A \Omega A^T \end{align*}
$Y \sim \mathcal{N} \left( A \mu_X, A \Omega A^T \right) $
Throughout, the assumption is that $A$ is a deterministic matrix and thus, can be pulled out of the expectation operator.
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