# Another jpoint distribution problem

• Apr 14th 2007, 08:10 AM
CaptainBlack
Another jpoint distribution problem
Quote:

problem 2:
A traffic engineer is studying the number of vehicles that arrive, during a certain 2-minute period, at two streets corners that are close to each other. Let X be the number of vehicles at on street corner and Y the number at the other. The enginner knows the joint distribution probability distribution of X and Y is:
-f(x,y)=(9/16)*(1/4)^(x+y), x=0,1,2,3..., y=0,1,2,....
- 0 elsewhere

For example P(X=1,Y=1)=0.035.
The enginner calculated E(X)=E(Y)=1/3, and Var(X)=Var(Y)=4/9.
He has calculated that X and Y are independent.

I'm not sure what is being asked here. The X and Y are independent
can be deduced form:

f(x,y)= g(x)g(y), x=0,1,.. y=0,1, ..

where g(x)=(3/4)(1/4)^x since g(x) is a probability distribution, and as
the joint distribution is the product of marginal distributions for X and Y,
X and Y are independent.

Quote:

1-Find rho, the correlation coefficient of X and Y.
From the book, I get the formula
rho=E(((X-mean1)/var1)*((X-mean2)/var2))
but when I tried to compute it, I get undefined.
rho= sum_{x=1 to infty, y=1 to infty} (x-xbar)(y-ybar)/sqrt(var(x)var(y)) f(x,y)

but this sum is seperable to:

rho= [sum_{x=1 to infty} (x-xbar)/sqrt(var(x)) g(x)][sum_{y=1 to infty} (y-ybar)/sqrt(var(y)) g(y)] = 0

RonL
• Apr 14th 2007, 04:24 PM