Correlation and independence in Probability

[vikkisut88]

Homework Statement

Let X be a random vairable which can only take three values: -1,0,1 and they each have the same probability. Let Y also be a random vairable defined by Y = X². Show that
i) X and Y are not independent
ii) X and Y are uncorrelated

Homework Equations

To show that two random variables are independent, you show that E(XY) /= E(X)E(Y)
To show that two random variables are uncorrelated, you show that $$\rho$$_x,y = 0

The Attempt at a Solution

I have found E(X) = 0 and E(Y) = 2/3 and so E(X)E(Y) = 0, however i don't know how to calculate E(XY) as I don't know how to construct the joint probability mass function table for X and Y

 

[e(ho0n3]

Why are you dealing with expected values? Why not just deal with the prob. mass functions, i.e. X and Y are independent if p(x, y) = p_X(x) p_Y(y).

Determining p(x, y) is straightforward. For example, p(0, 1) = P{X = 0 and Y = 1} = P{X = 0 and X² = 1} = P{X = 0} P{X² = 1 | X = 0} = 0 since P{X² = 1 | X = 0} = 0, which I hope you'll agree with.

 

[vikkisut88]

Okay so I'll work with pmf's instead, but then for example finding P{X=0 and Y=0} = P{X=0 and X²=0} = P{X=0}P{X²=0|X=0} would that be equal to 1/3 * 1 as P{X²=0|X=0} that is one?

 

[e(ho0n3]

Yes. What else could it be?

 

[rochfor1]

In fact, since they are uncorrelated (at least you want to show that) you expect that E(XY)=E(X)E(Y).

 

[vikkisut88]

exactly and that's why I'm really confused...i think this must be a special case or something.

 

[rochfor1]

I think you're making a very common error by assuming that two random variables are uncorrelated if and only if they are independent. This statement is in fact not true, as this problem illustrates. For arbitrary random variables, X and Y independent implies uncorrelated, but uncorrelated does not necessarily imply independent, unless further assumptions are made. An assumption that makes this implication true is that X and Y are Gaussian, which is not true for the variables you are given.

 

[rochfor1]

Which is why as e(ho0n3 said, you have do deal with the joint pmf and not the expectations to show that X and Y are not independent.

 

[vikkisut88]

but that is what i indeed went away and did and this gave me that E(XY) = E(X)E(Y) which implies independence. That is the only way i know how to/have been taught how to work out whether two random variables are independent or not. Therefore is there another way?

 

[rochfor1]

Like I said, that doesn't necessarily show that the RVs are independent, as this problem illustrates. If X and Y are discrete random variables (like in this problem), to show that they're independent, show that P( X = x, Y = y ) = P( X = x ) P( Y = y ).

 

[vikkisut88]

ah okay - thank you! i had misread what had been previously typed about that, i do apologise