Bivariate expected value and variance

ArcanaNoir

Homework Statement

I need to know these formulas to answer the homework problems, but I can't squeeze the forumlas out of the gibberish in the book, so I'm asking for varification of the formulas.

For a bivariate probablity density function, for example f(x,y)= 2xy when x and y are between 0 and 3, and 0 elsewhere,

expected value of x: E[X]
expected value of (X,Y)
Variance of x: Var[X]
Var[X,Y]

$$E[x] = \int_{-\infty }^{\infty }xf_1(x) \: \mathrm{d}x$$ where $f_1(x)$ is the marginal probability distribution of x. Is this correct?

Now, for E[X,Y], do you think the book means the expected value of the product XY? because the only formula it gives here is for the product. So, E[XY]. If they really mean E[X,Y] and not E[XY], then is there a formula for E[X,Y]? I don't have one in my book.

As for Var[X], is it $$Var[x] = \int_{-\infty }^{\infty }x^2f_1(x) \: \mathrm{d}x -(E[x])^2$$ ?

And for Var[X,Y], I have no idea, the only formulas I see in my book are for Var[X+Y].

Remember, these are all for bivariate distributions.