Homework Help: Show that if X is a bounded random variable, then E(X) exists.

1. Jan 24, 2012

number0

1. The problem statement, all variables and given/known data

Show that if X is a bounded random variable, then E(X) exists.

2. Relevant equations

3. The attempt at a solution

I am having trouble of finding out where to begin this proof.

This is what I got so far:

Suppose X is bounded. Then there exists two numbers a and b such that P(X > b) = 0 and P(X < a) = 0 and P(a <= X <= b) = 1.

I have no idea if I am even doing this right. Anyone wanting to take a crack at this one? Thanks.

Last edited: Jan 24, 2012
2. Jan 25, 2012

Stephen Tashi

How do your course materials define a "bounded random variable"? The claim "if X is a bounded random variable, then E(X) exists" isn't true using the usual definition of a "bounded random variable".

Perhaps your problem assumes the domain of X is bounded as well as the range.

3. Jan 25, 2012

number0

The book specifically defined X as bounded as the following:

|X| < M < ∞ .

Here is the whole question, word for word:

Show that if a random variable is bounded—that is, |X| < M < ∞—then
E(X) exists.

I do not know about the range though.

Last edited: Jan 25, 2012
4. Jan 25, 2012

Stephen Tashi

I see what the book is doing now. I was thinking of a probability density function being "bounded" as meaning it is a function with a bounded range on a possibly infinite domain. Your book means the domain is bounded and you probably get to assume the range of the function is [0,1] or a subset of it. So you can bound the integral of the expression x f(x) by (M)(1) = M.

Have you studied theorems that say something to the effect that bounded (in range) continuous function on a closed interval is integrable?

5. Jan 25, 2012