S(t) = a0 + a1X(t) + a2X2(t)
where X(t) is the random displacement, a Gaussian random process, and stationary.
1) determine the PDF of S(t)
2) determine the joint PDF of stress and stress velocity, S(t) and S'(t).
3) how would you determine the PDF of S'(t)?
for 1) PX(x) = 1/sqrt(2∏σ2) exp [-(x-μ)2/2σ2]
where μ= ∫xPX(x)dx
for 2) ??
for 3) extract the PDF of S'(t) from the joint PDF setting one of the bounds of S(t) to ∞.
The Attempt at a Solution
For 2) I was going to get the joint PDF by using a linear combination of the PDF of each through computing the mean, the variance and the standard deviation. If I assume independence then I could use PS(t)S'(t) = PS(t)PS'(t) and that would lead me into a circle with 3). If anyone can make a suggestion how to approach the problem I would be greatly appreciative!!