We assume that for a non-negative, bounded, continuous random variable we have $$ E[X]=\int_0^\infty P(X>x) dx $$ Now the task is to extend this result to non-negative, continuous random variables by using the Monotone Convergence Theorem.

I am not sure how one achieves this. I assume one has to show that for a positive, continuous rv $X$, one can construct a series $\lbrace X_n \rbrace$ of bounded non-negative countinuous rv's such that $X_n \uparrow X$, but I am not sure how to that or what result can be used. I guess my problem is finding out when a random variable is the limit of a sequence of random variables.

up vote 1 down vote accepted

Hint: consider the sequence of random variables $X_n = X\wedge n$ for $n\in\mathbf{N}$, where $a\wedge b = \min\{a,b\}$.

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