Random vibrations/engineering probability & statistics

1. Apr 7, 2012

L.Richter

1. The problem statement, all variables and given/known data

Stress :
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)?

2. Relevant equations
for 1) PX(x) = 1/sqrt(2∏σ2) exp [-(x-μ)2/2σ2]
where μ= ∫xPX(x)dx
σ2= ∫(x-μ)2PX(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 ∞.

3. 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!!
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

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