$X$ and $Y$ have bivariate normal distribution with zero means, unit variances, and correlation $\rho$. Let $U$ and $V$ be independent of $X$ and $Y$.

How can we find the distribution of $Z = \displaystyle\frac{UX + VY}{\sqrt{U^2 + 2\rho UV + V^2}}$

I was thinking of using the characteristic function by considering the characteristic function of $Z$, but I don't really know how to proceed.

  • $\begingroup$ What are $U$ and $V$? Are they independent from or correlated to each other? What are their distributions? $\endgroup$ – hypernova Apr 25 '18 at 11:52
  • $\begingroup$ $U$ and $V$ are random variables. We know anything about their distributions. $\endgroup$ – sedrick Apr 25 '18 at 12:06
  • $\begingroup$ May I ask where did you find this problem? $\endgroup$ – StubbornAtom Apr 27 '18 at 11:48
  • $\begingroup$ @StubbornAtom This was from Chapter 5 of Grimmett and Stirzaker's Probability and Random Processes $\endgroup$ – sedrick Apr 29 '18 at 5:31

From the fact that distribution of linear combination of jointly normal variables is univariate normal, one can show that $uX+vY$ is normally distributed, where $u,v$ are real constants. You can verify this using moment generating functions. The MGF of $uX+vY$ is given by

\begin{align} M(t)&=E(\exp(tuX+tvY)) \\&=E[\exp(tuX)E(\exp(tvY)\mid X)] \\&=E[\exp(tuX)\exp(\rho tv X+(1-\rho^2)t^2v^2/2)] \\&=\exp(1-\rho^2)t^2v^2/2)E(\exp((tu+tv\rho)X)) \\&=\exp((1-\rho^2)t^2v^2/2)\exp((tu+tv\rho)^2/2) \\&=\exp(t^2(v^2+u^2+2uv\rho)/2) \end{align}

, which is the MGF of a $\mathcal{N}(0,u^2+v^2+2uv\rho)$ distribution.

Known facts used above are that $Y\mid X\sim\mathcal N(\rho X,1-\rho^2)$ and the MGF of a univariate normal variable. Also note that $X$ is standard normal.

By uniqueness of MGF, we conclude that $$uX+vY\sim\mathcal{N}(0,u^2+v^2+2uv\rho)$$

That is, $$Z\mid (U=u,V=v)=\frac{uX+vY}{\sqrt{u^2+2\rho uv+v^2}}\sim\mathcal{N}(0,1)$$

So the distribution of $Z$ conditioned on $U=u$ and $V=v$ is independent of $u$ and $v$. This means the unconditional distribution would be the same as the conditional distribution.

Hence, $$Z=\frac{UX+VY}{\sqrt{U^2+2\rho UV+V^2}}\sim\mathcal{N}(0,1)$$

A direct solution is also possible from the MGF or the characteristic function of $Z$, conditioned on $U$ and $V$.


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