# Prob and stat expected value of x

1. Jan 2, 2009

### Proggy99

1. The problem statement, all variables and given/known data
The distribution function of a random variable X is given by

F(x) = {
0 if x < -3
3/8 if -3 $$\leq$$ x < 0
1/2 if 0 $$\leq$$ x < 3
3/4 if 3 $$\leq$$ x < 4
1 if x $$\geq$$ 4

Calculate E(X), E(X$$^{2}$$ - 2|X|), E(X|X|).

2. Relevant equations

3. The attempt at a solution
I actually have no idea how to start this. None of the chapter examples seem to give me a clue on what to do. I have calcuated E(X) for probability mass function equations, but not distributive functions like the above. Can someone give me a strong hint on how to approach this for E(X) and I can take it from there? Thank you for any help.

And sorry for the formatting, I could not get it to look right in latex, but I think it should be understandable.

2. Jan 2, 2009

Well, what is the definition of E(X)? Further on, what is the relation between F(X) and f(x) (i.e. the probability density function f of the random variable X)?

3. Jan 2, 2009

Okay, another hint. Since you need to find f(x) to calculate E(X), is there a way to calculate f(x) from F(x)? [Hint 2: f(xi) = F(xi) - F(xi-), where F(xi-) is the left limit of F when x --> xi form the left.]

4. Jan 2, 2009

### Proggy99

Well, E(X) is the expected value of X. E(X) = $$\sumxp(x)$$ where p(x) is the probability mass function. My main issue with the problem is that I am confused by the lack of an equation...

okay, so I started typing that and now I am wondering if it is as simple as this:
-3 * 3/8 + 0 * 1/8 + 3 * 2/8 + 4 * 2/8 = 5/8

I only looked at the numbers where it jumped on a graph and I looked at actual chances of each happening. In other words I subtracted f(x2) - f(x1) to get the chances of f(x2). I know the answer is 5/8, am I doing this correct or did I just get a coincidentally equal number?

5. Jan 2, 2009

Yes, my result is 5/8 too. So, the random variable is given with $$X = $\left( \begin{array}{cccc}-3 & 0 & 3 & 4 \\ 3/8 & 1/8 & 1/4 & 1/4 \end{array} \right)$$$. If you "calculate" F(x) from X, you'll get your given function F.

6. Jan 2, 2009

### Proggy99

so then what I really mean to say would be:
-3 * (3/8 - 0) + 0 * (1/2 - 3/8) + 3 * (3/4 - 1/2) + 4 * (1 - 3/4) = 5/8

technically the same thing, but clearer in showing what I was doing

and for E(X - 2|X|) would I then plug in -3 and replace the -3 above with the solution? such as (-3$$^{2}$$ - 2|-3|) = 3
so the equation above would start as 3 * (3/8 - 0) + ... ?
*edit*dropped the square, fixed now

7. Jan 2, 2009

### Proggy99

answered my own question by working it out to find that I got the correct answer. Thanks for the hints radou, it is making sense now!

8. Jan 2, 2009