Need help in Engineering Statistics/Random Vibrations

L.Richter

Stress given as:

S(t) = a0 + a1X(t) + a2X^2(t)

where X(t) is the random displacement, Gaussian random process and is stationary.

Determine the joint probability density function of the stress and stress velocity, S(t) and S'(t). I was considering using the mean, the variance and the standard deviation of both to find their respective probability density functions (PDF) and then using a linear combination of them to complete the joint PDF. But, the next question is how would you extract the PDF for S'(t) from the joint PDF. That would take me in a circle.

Any one out there have an idea on how to approach the problem? I would greatly appreciate any suggestions!