Covariance of two dependent variables

[jegues]

Homework Statement

See figure attached

Homework Equations

The Attempt at a Solution

I am not concerned with part (a), I have deduced that indeed X and Y are dependent.

I'm not sure if I have done part (b) correctly, and I am quite certain I have done part (c) incorrectly, but I couldn't think of what else to do.

Am I on the right track in parts (b) and (c)? What is incorrect, or how should I approach the problem?

Thanks again!

 

[jegues]

Here is the last page of my attempt.

 

[haruspex]

B) looks ok. you should expect zero covariance. The two are clearly anticorrelated where X is negative and, equally, correlated where X is positive.
In c), something strange at the bottom of the first sheet. How did μ_Xμ_Y become X?

 

[jegues]

B) looks ok. you should expect zero covariance. The two are clearly anticorrelated where X is negative and, equally, correlated where X is positive.
In c), something strange at the bottom of the first sheet. How did μ_Xμ_Y become X?

I calculated μx for that case, it was found to be 1.

Since Y = |X| + N, and X is in the range of (0,2) |X| = X.

Thus, Y = X + N, I figured this pdf would simply be the pdf of N (it is known to be Gaussian) shifted to the right by X, having a mean value of X. Hence, μy = X.

I don't think that is correct, because I think μy should be a number.

Where did I go wrong? How do I fix part (c)?

 

[haruspex]

I don't think that is correct, because I think μy should be a number.

Quite so. E[Y] = E[|X|+N] = E[|X|]+E[N].

 

[jegues]

Quite so. E[Y] = E[|X|+N] = E[|X|]+E[N].

Okay so from this,

$$E[Y] = E[X] + 0 = \mu_{X} = 1 = \mu_{Y}$$

but I am still stuck in finding,

$$\sigma_{XY} = E[(X-\mu_{X})(Y-\mu_{Y})] = E[(X-1)(Y-1)] = E[(X-1)(|X|+N-1)]$$

Any ideas?