Find the probability density $f_Y(y)(Y= X_1 + X_2 + 1)$

Question

Problem: Let $X=[X_1,X_2]^T$ be a real-valued Gaussian random vector with mean and co-variance

$$\mu:=[1, 2]^T\,\,\,,\,\,\, \Lambda := \begin{pmatrix}2&-1\\-1&3\end{pmatrix}$$

Find the probability density $Y = AX + B$, where $A := [1, 1]$ and $B := 1$

What I did: I acquired the probability density of $X$, which is $$f_X(x)=\frac{1}{2\sqrt{5}\pi}\exp\left\{-\frac{3x_1^2+2x_2^2-10x_1-10x_2+2x_1x_2+15}{10}\right \}$$ (hopefully this form is correct). Since I was asked to find the probability density of $Y=X_1+X_2+1$, I tried to transform the $3x_1^2+2x_2^2-10x_1-10x_2+2x_1x_2+15$ part into a polynomial including multiple $(x_1+x_2+1)$ terms. Even though this sounds fair, it is not easy to find such a polynomial.

https://en.m.wikipedia.org/wiki/Multivariate_normal_distribution see the part of affine transformation — vita nova, Oct 08 '17 at 22:38

score 1 · Answer 1 · answered Oct 08 '17 at 23:05

1

If $X\sim N(\mu,\Sigma)$ where $\mu\in\mathbb R^{a\times1}$ and $\Sigma\in\mathbb R^{a\times a}$ and $A\in\mathbb R^{b\times a}$ then

$$ AX+B \sim N( A\mu+B, A\Sigma A^T). $$

I'd do that first and then afterwords think about the density.

answered Oct 08 '17 at 23:05

Michael Hardy

1

Yarp, but you should also add full rank condition for $A$. – Jirapat Samranvedhya Oct 08 '17 at 23:20
@JirapatSamranvedhya : "add[ing] a full rank condition" can be regarded as part of "afterwards think[ing] about the density". At any rate, in this case, the given matrix has full rank. – Michael Hardy Oct 09 '17 at 00:28

Find the probability density $f_Y(y)(Y= X_1 + X_2 + 1)$

1 Answers1