# Probability distribution of two variables

1. Apr 7, 2013

### mjordan2nd

1. The problem statement, all variables and given/known data

Consider a Hamiltonian involving two Gaussian variables, X and Y. Start from the statement that the average formed by these two variables is of the form

$$<e^{aX+bY}> = e^{a^2+b^2-ab}$$

2. Relevant equations
$$<e^{ax}> = \int_{-\infty}^{\infty} dx \frac{exp(-\beta ( x-<X>)^2/2)}{\sqrt{2 \pi / \beta}} e^{ax}$$

3. The attempt at a solution

Since X and Y are Gaussian the joint probability is P(x,y)=P(x)P(y) -- so we have two integrals. Using the definition above I found

$$<e^{ax}> = \sqrt{e^{\beta_x \left[ \left( <x> + \frac{a}{\beta_x} \right)^2 + <x> \right]}}$$
$$<e^{by}> = \sqrt{e^{\beta_y \left[ \left( <y> + \frac{b}{\beta_y} \right)^2 + <y> \right]}}$$

We can also say <exp[ax+by]> = <exp[ax]><exp[by]>, so combining the two above we find the desired expectation value. I'm not sure how to proceed from here to calculate the expectation values and the variances of the variables though. Any help would be appreciated. Thanks.

Last edited: Apr 7, 2013