correlation co-efficient of a joint pdf
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From: rabbits77 on 14 Dec 2009 13:10 Let X and Y have the joint pdf f(x,y)=x+y 0<=x<=1, 0<=y<=1 =0 Elsewhere How do I find the correlation co-efficent between X and Y? Actually, I think my main problem is calculating the Var(X) and Var(Y). I keep getting these very messy expressions that simply can't be right. How do I calculate Var(X) and Var(Y) here?
From: Robert Israel on 14 Dec 2009 13:56 rabbits77 <rabbits77(a)my-deja.com> writes: > Let X and Y have the joint pdf > > f(x,y)=x+y 0<=x<=1, 0<=y<=1 > =0 Elsewhere > > How do I find the correlation co-efficent between > X and Y? > Actually, I think my main problem is calculating > the Var(X) and Var(Y). > I keep getting these very messy expressions that > simply can't be right. > How do I calculate Var(X) and Var(Y) here? For example, E[X^2] = int_0^1 int_0^1 x^2 (x+y) dx dy E[X] = int_0^1 int_0^1 x (x+y) dx dy Var(X) = E[X^2] - E[X]^2 These should not be very messy. -- Robert Israel israel(a)math.MyUniversitysInitials.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada
From: Ray Vickson on 14 Dec 2009 18:45
On Dec 14, 10:10 am, rabbits77 <rabbit...(a)my-deja.com> wrote: > Let X and Y have the joint pdf > > f(x,y)=x+y 0<=x<=1, 0<=y<=1 > =0 Elsewhere > > How do I find the correlation co-efficent between > X and Y? > Actually, I think my main problem is calculating > the Var(X) and Var(Y). > I keep getting these very messy expressions that > simply can't be right. > How do I calculate Var(X) and Var(Y) here? There are two ways: (1) Directly, as Robert Israel has shown; or (2) get the marginal one-dimensional density of X (or Y) and then use that. (Here, we can exploit the fact that to get EX or Var(X), etc., we just need the marginal distribution of X.) The marginal density f_X of X is given by f_X(x) = int{y=0..1} f(x,y) dy, which is pretty easy to do and is not at all messy. R.G. Vickson |