# Stochastic Processes

1. Feb 16, 2008

### wildman

1. The problem statement, all variables and given/known data
I need someone to reassure me (or correct me) on this problem:

The process $$X(t) = e^{At}$$ is a family of exponentials depending on the random variable A.
Express the mean $$\eta(t)$$, the autocorrelation $$R(t_1,t_2)$$, and the first order density f(x,t) of X(t) in terms of the density $$f_a(a) of A$$

2. Relevant equations

$$f(x,t) = \frac {\partial F(x,t)} {\partial x}$$
$$\eta (t) = \int_{-\infty}^{\infty} xf(x,t)dx$$
$$R(t_1,t_2) = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} x_1 x_2$$
$$f(x_1,x_2;t_1,t_2)dx_1 dx_2$$

3. The attempt at a solution

$$f_a(a) = ae^a$$
$$f(x,t) = ae^{at}$$
$$\eta(t) = \int_{-\infty}^{\infty} a^2 e^{at} da$$
$$R(t_1,t_2) = \int_{-\infty}^{\infty} a^4 e^{at_1} e^{at_2} da$$

Last edited: Feb 16, 2008