0

Let $X = AY + Z$ where $A$ is constant, $Y$ and $Z$ are independent gaussian rvs. How do you prove that $X$ and $Y$ are jointly gaussian? I know that the sum of two independent gaussians gives a gaussian distribution. But I have no idea where to start to show joint gaussian pdf.

eet
  • 393

1 Answers1

1

For every linear combination of $X$ and $Y$ : $$ \mu X + \beta Y = (\beta + \mu A) Y + \mu A Z$$ And since $Y$ and $Z$ are independent, the linear combination follows a gaussian distribution. So they are jointly gaussian.

purple
  • 583
  • How does linear combination of two gaussians imply the joint gaussianity? – eet Jan 10 '21 at 16:48
  • By definition. $X$ and $Y$ are jointly gaussian if every linear combinations are gaussian. – purple Jan 10 '21 at 16:51
  • Can you give me a reference about this? – eet Jan 10 '21 at 16:52
  • https://en.wikipedia.org/wiki/Multivariate_normal_distribution#Equivalent_definitions Moreover, the equivalence is straighforward : think about the characteristic function – purple Jan 10 '21 at 17:02