# On the expected value of a random integral:

Is it possible to find the expected value of $u(t)$ in terms of the following information: $$u(t)=\int_{0}^{t}(t-s)(f(s)+(T-s)Y)X_sds$$ where:

$X_s$ is a wide sense stationary process with known probability density function, statistical moments, and (higher order) spectral density function(s).
$f(t)$ is a known real function.
$T>0$ is a constant and $0<t<T$.
$Y$ is an unknown random variable which satisfies $u(T)=0$.

• Fubini's theorem? Your $ds$ integral and the expectation are both integrals... Jan 3, 2014 at 7:56

you can just eliminate $Y$ from the equation for $u(t)$, by means of
$$Y=-\frac{\int_0^T (T-s')f(s')X_{s'} ds'}{\int_0^T (T-s')^2 X_{s'} ds'}$$
then the probability distribution of $u(t)$ is given entirely in terms of the stochastic properties of $X_s$, so the answer to your question is "yes", it is possible. However, since the dependence of $u(t)$ on $X_s$ is highly nonlinear, there is no simple general expression for the expectation value.
• Thanks. It seems that you got the point very clearly. The problem is with $Y$. I have tried to write $Y$ with $X_s$ appearing only in numerators (Using Fubini's theorem and its corollary for integration of convolutions, and thereof solving an integral equation of the first kind which is ill-posed of course leading to a complicated result). But what do you suggest for this highly nonlinear problem? What should I try more? Jan 3, 2014 at 10:07
• I don't think any progress can be made without further input on the stochastic properties of $X_s$. Jan 3, 2014 at 10:29