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?
Best Answer
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.)