I have an integral I need to evaluate that contains a random variable? How would I go about doing something like that. For instance lets say I have a random variable $C$ with PDF $f_C(x)=1/N \; 0\le x\le N$ (Uniform distribution) and I have an integral $$ \int\limits_a^bC\mathrm{d}x $$

Can I evaluate an integral like the above? What if the integral was: $$ \int\limits_a^bCf(x)\mathrm{d}x $$ Could I evaluate that integral? If yes, then how?

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    $\begingroup$ The intent of your question is not completely clear to me. You define the density of $C$ over the same variable you're taking the integral over. Did you mean to integrate the density with respect to $dx$? If on the other hand you're integrating the random variable rather than its density, is $C$ a constant (but random) value over the range of $x$, or is it a different, independent, uniform value at every value of $x$ in $(a,b)$? Some context might help. $\endgroup$ – Glen_b Sep 3 '13 at 2:02
  • $\begingroup$ Ohh okay I think I realize what your saying... now that I've taken a couple minutes off I see that the question actually doesn't make much sense C doesn't have to be inside the integral. It doesn't depend on x. $\endgroup$ – Twiltie Sep 3 '13 at 2:32

Doesn't make much sense. Remember that $C$ is a (measurable) map from $\Omega$ (the underlying sample space) to $\mathbb{R}$. Hence, strictly speaking $$ \int_a^b C \,dx = \int_a^b C(\omega) \,dx = C(\omega) \int_a^b \,dx = C(\omega) \cdot (b-a) \, . $$

(P.S. When you have a stochastic process $Z:\Omega\times\mathbb{R}\to\mathbb{R}$, then under certain conditions $I:\Omega\to\mathbb{R}$ defined by $$ I(\omega) = \int_a^b Z(\omega,x)\,dx $$ is a random variable. For details, take a look at the classic probability books by Loève, Neuveu and Doob.)


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