Determining joint probability density function

[L.Richter]

Homework Statement

A process X(t) is defined as

X(t) = Asin(ωt + $\phi$)

where A and $\phi$ are random variables while ω is a deterministic parameter. Note that 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

PDF of X(t), P_X(t)(x) = ∫F_X(x)dx

Is there another way to determine the PDF of a function that is multivariate, but not necessarily exponential, Gaussian or Gamma?

The Attempt at a Solution

I am not understanding how to get a joint PDF in terms of another joint PDF. Please advise. I'm not sure which functions belong where in the equations. I put the PDF equation since its the one I'm struggling with. I know that if I assume independence then I can get the joint PDF as a product of the individual PDFs. Thanks in advance!

 

[mighty2000]

There are a few different ways you could approach this problem. One approach would be to use the transformation method, where you first find the joint PDF of A and ϕ and then use that to find the joint PDF of X(t) and X'(t). This method would require you to use the Jacobian determinant to convert between the two sets of variables.

Another approach would be to use the definition of a joint PDF, which is the probability of the random variables taking on a specific set of values. In this case, you would need to determine the probability that X(t) and X'(t) take on specific values, given the joint PDF of A and ϕ.

It's also worth noting that since A and ϕ are random variables, the joint PDF of X(t) and X'(t) will also be a random variable. So even if you know the joint PDF of A and ϕ, you will still need to take into account the variability in those variables when determining the joint PDF of X(t) and X'(t).