Discussion between Jac Frall and nomen

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18:22

Q: Find $E[XY]$ given that $X \sim N(0,1)$ and $Y$ is a RV such that $E[Y|X=x] = ax+b$ and $Var [Y|X=x]=1.$

Let $X \sim N(0,1)$ and let $Y$ be a RV such that $E[Y|X=x] = ax+b$ and $Var [Y|X=x]=1.$ Find $E[XY]$. I found $E[Y]$ using the law of total expectation, and I found $Var[Y]$ using the law of total variance. Now I am not sure how to proceed since I do not know the joint density function ...

nomen

Y is a linear function of X, so it is very much not independent. But there are formulas for this situation. (Not that I remember them at the moment, but perhaps they're in whatever book you're using).

Jac Frall

@nomen That is what I thought. I've gone through the book and can't find anything relevant to this scenario.

nomen

do you have any formulas to calculate the covariance between X and Y? If so, look at the formula that defines C(X,Y) carefully and see what you can figure out based on the problem data.

Jac Frall

@nomen The formulas I have for Covariance that would allow me to solve it require me knowing what the value of the Covariance is. Is this not the case with the formula you have?

nomen

That depends. In this situation, (X,Y) are jointly normal, so there is a simple formula for the correlation and covariance. But if your book didn't cover it, you'll have to prove the formula.

Jac Frall

18:22

@nomen how do we know that the RVs are jointly normal?

nomen

Because Y is a linear function of X, so Y is normal.

Jac Frall

Ok great. I have a lot of formulas and am looking through my textbook now. I also wonder if this would work

Product distribution

A product distribution is a probability distribution constructed as the distribution of the product of random variables having two other known distributions. Given two statistically independent random variables X and Y, the distribution of the random variable Z that is formed as the product Z = X Y {\displaystyle Z=XY} is a product distribution. == Algebra of random variables == The product is one type of algebra for random variables: Related to the product distribution are the ratio distribution, sum distribution (see Lis...

^ it is linked to the relevant part

nomen

I think the integral would be hard.

By the way, I was slightly wrong -- Y isn't necessarily normal, BUT X and Y fit the axioms for the standard linear model, so the correlation is a...

You basically want to do an exact version of en.wikipedia.org/wiki/Simple_linear_regression (which is the statistical "sample" based argument)

Jac Frall

E[XY] = E[E[XY|X]] = E[X*E[Y|X] = E[X*(aX+b)] = 0
This is what I get using the formula from wikipedia about the expectation of the product

Seems like there is no integral required. I likely did it wrong though

I feel like there must be another way. This is a theory based class so the Linear regression assumptions probably arent valid for this problem

2 hours later…

gd1035

20:49

How are you getting the last expectation to be 0?

E[X(aX+b)]=E[aX^2+bX]=aE[X^2]+bE[X]=aE[X^2]

Do you recognize the distribution of X^2?

nomen

21:24

If you're still working on this, look up 18.3.1 at stat.rutgers.edu/home/hcrane/Teaching/582/lectures/…, which explains (well) what I was getting at (poorly).

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