Prob and stat expected value of x

[Proggy99]

Homework Statement

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|).

Homework Equations

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.

 

[radou]

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)?

 

[radou]

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.]

 

[Proggy99]

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)?

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?

 

[radou]

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

 

[Proggy99]

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.]

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

 

[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!

 

[radou]

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!

No problemo! It's nice to hear when someone works the answers out alone.