# Distribution of $XY$ when $(X,Y) \sim BVN(0,0,1,1,\rho)$

The question is pretty much in the title,

I need to find an approximate distribution of $$XY$$ when $$(X,Y)$$ follow a Bivariate Normal Distribution where $$X$$ and $$Y$$ are each $$N(0,1)$$ distributed and $$Cov(X,Y) = \rho.$$

I proceeded to find the required using transformation of variables, but I am stuck at the following integration,

$$\displaystyle \int_{-\infty}^{\infty} \left| \frac{1}{u}\right| \text{exp} \left\{ \frac{-1}{2(1-\rho^2)}\left(u^2 + \frac{v^2}{u^2}\right)\right\}\,du$$

How do I proceed to solve this integral? Or is there any other way obtaining the required distribution?

This problem is however an intermediate step of another question that I am trying to solve,

Suppose $${(X_1, Y_1), . . . ,(X_n, Y_n)}$$ is a random sample from a bivariate normal distribution with $$E(X_i) = E(Y_i) = 0, \, Var(X_i) = Var(Y_i) = 1$$ and unknown $$Corr(X_i, Y_i) = \rho \in (−1, 1),$$ for all $$i = 1, . . . , n.$$ Define $$W_n = \frac{1}{n}\sum_{i = 1}^n X_iY_i$$. For large $$n$$, obtain an approximate level $$(1 − \alpha)$$ two-sided confidence interval for $$\rho$$, where $$0 < \alpha < 1.$$

Though, I know about the Fisher's z transform, and its asymptotic distribution. But I think the question requires me to give an answer in terms of $$W_n$$ only. How should I solve this?

Can I use CLT in this? Because I ran a simulation for 1000 times and the distribution of $$W_n$$ came out to be convincingly normal.

• pretty sure that is a inverse Gaussian pdf/kernel Commented Apr 26, 2019 at 5:17
• It doesn't looks like Inverse Gaussian @probabilityislogic. The integral isn't the same as the PDF mentioned here Commented Apr 26, 2019 at 7:42
• After confirming the integral converges, merely observe that the integrand is an odd function of $u$ to conclude the integral must be zero.
– whuber
Commented Apr 26, 2019 at 13:45
• sorry I meant the generalised version here en.m.wikipedia.org/wiki/… Commented Apr 26, 2019 at 14:27
• In that case the integral is proportional to a Bessel function $2K_0(v/(1-\rho^2)).$
– whuber
Commented Apr 26, 2019 at 15:00

## 2 Answers

Your original question does not require you to find the distribution of $$XY$$ when $$(X,Y)$$ is jointly normal. Here is a hint for that question:

Let $$Z_i=X_iY_i$$, so that $$Z_1,Z_2,\ldots,Z_n$$ are i.i.d variables. Hence by classical CLT we have

$$\frac{\sqrt n(W_n-\operatorname E(Z_1))}{\sqrt{\operatorname{Var}(Z_1)}}\stackrel{L}\longrightarrow N(0,1)\tag{*}$$

, where $$W_n=\frac{1}{n} \sum\limits_{i=1}^n Z_i$$.

Since we know the conditional distributions of a bivariate normal distribution, use law of total expectation to find $$\operatorname E(Z_1)$$:

$$\operatorname E(Z_1)=\operatorname E\left[\operatorname E(X_1Y_1\mid Y_1)\right]=\operatorname E\left[Y_1 \operatorname E(X_1\mid Y_1)\right]$$

Do this similarly for $$\operatorname E(Z_1^2)$$ and hence find $$\operatorname{Var}(Z_1)=\operatorname{E}(Z_1^2)-[\operatorname{E}(Z_1)]^2$$

Now $$(*)$$ actually gives you a pivot for constructing an asymptotic confidence interval for $$\rho$$.

• Thanks for the answer @StubbornAtom. I was a bit skeptical about using CLT at first but when I ran the simulation I was convinced. Apart from this answer, is there any way I could solve for the distribution(exact) of $XY$? Commented Apr 26, 2019 at 8:33
• I do not know of any standard result for the distribution of $XY$ when $X$ and $Y$ are correlated Normals. Commented Apr 26, 2019 at 8:35

An exact answer is given in Probability Distributions Involving Gaussian Random Variables: A Handbook for Engineers, Scientists and Mathematicians (along with many other results.)

If $$(X,Y)$$ is bivariate normal distributed with zero means and correlation $$\rho$$, then the density function of the product $$XY$$ is given by $$f_{XY}(z)= \frac1{\pi \sigma_1 \sigma_2} \exp\left\{\frac{\rho z}{\sigma_1 \sigma_2 (1-\rho^2)}\right\} K_0\left( \frac{|z|}{\sigma_1\sigma_2 (1-\rho^2)} \right)$$ where $$K_0$$ is the modified Bessel function of the second kind.

EDIT   answer for additional question in comment


The reference book given gives the characteristic function. For the moment generating function I get $$M_{XY}(t)= \left(1 - 2 t \rho - t^2(1-\rho^2)\right)^{-\frac{1}{2}}$$ for $$-1 (so you missed on $$t$$.)

• Thank you for this answer. Actually I was just required to find the approximate confidence interval, but I was a bit curious about the exact distribution. Now, I haven't seen this pdf before and nor I am familiar yet with the Bessel function. However, I tried to derive the MGF of $XY$ and it came out to be, $\left(1 - 2\rho - t^2(1-\rho^2)\right)^\frac{-1}{2}$. Could you verify if this is correct? Commented Apr 28, 2019 at 9:06
• Oh yes, I am so sorry. Must have missed while typing. Thanks for the clarification. Commented Apr 29, 2019 at 11:28