# Determining the joint probability density function

1. Apr 11, 2012

### L.Richter

1. The problem statement, all variables and given/known data
A process is defined as:

X(t) = Asin(ωt+$\phi$])

where A and $\phi$are random variables and ω is deterministic. A is a positive random variable.

Determine the joint probability density function, PDF, of X(t) and X'(t) in terms of the joint PDF of A and$\phi$.

2. Relevant equations

PA(a) = ∫a FA$\phi$(a,$\varphi$)da
P$\phi$($\varphi$) = ∫$\phi$ FA$\phi$(a,$\varphi$)d$\varphi$

joint PDF = PA(a)P$\phi$($\varphi$)

joint PDF of X(t) and X'(t) ??

3. The attempt at a solution

I'm confused on how to get a joint PDF of functions X(t) and X'(t) out of a function of A and $\phi$.

Any suggestions would be greatly appreciated. It was suggested to assume there is a FA$\phi$(a,$\varphi$). But I'm still confused.

Last edited: Apr 11, 2012