Determining the joint probability density function

[L.Richter]

Homework Statement

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$\phi$.

Homework Equations

P_A(a) = ∫a F_A$\phi$(a,$\varphi$)da
P$\phi$($\varphi$) = ∫$\phi$ F_A$\phi$(a,$\varphi$)d$\varphi$

joint PDF = P_A(a)P_$\phi$($\varphi$)

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

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 F_A$\phi$(a,$\varphi$). But I'm still confused.

 

[mighty2000]

Thank you for your post. I can help you with your question. The joint probability density function (PDF) is a function that describes the probability of two or more random variables occurring simultaneously. In this case, we have two random variables, A and \phi, which are related to the functions X(t) and X'(t). The joint PDF of X(t) and X'(t) can be written as follows:

f(X(t), X'(t)) = PA(a)P\phi(\varphi)

Where PA(a) is the probability density function of A and P\phi(\varphi) is the probability density function of \phi. We can rewrite this as:

f(X(t), X'(t)) = ∫A ∫\phi FA\phi(a,\varphi)da d\varphi

This means that to find the joint PDF of X(t) and X'(t), we need to integrate the joint PDF of A and \phi with respect to A and \phi. This is possible because we are given the joint PDF of A and \phi, which is FA\phi(a,\varphi).

I hope this helps you understand how to get the joint PDF of X(t) and X'(t) in terms of the joint PDF of A and \phi. If you need further clarification, please don't hesitate to ask. Good luck with your research!