Suppose we have a continuous-time stochastic process $X(t)$, which consists of a sequence of delta functions that, at each time $t$, have a probability $p(t)$ of taking a non-zero value. $p(t)$ lies in $[0,1]$ for all $t$ and just a scaled survival function of an exponential distribution, so $p(t)$ is decreasing in time. We are interested in the random variable $Y = \int_0^\infty X(t) f(t) dt$ where $f(t)$ is a (deterministic) decreasing function of time. Is it possible to take the expectation of this integral; that is, $E\left[Y\right] = E\left[\int_0^\infty X(t) f(t) dt\right]$?
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