Determine the value of c and the covariance and correlation for the joint probability density function f(x, y) = c for 0< x < 5, 0 < y and x - 1 < y < x + 1.
I am confused on how to set-up for the bounds for the integration. It looks like you have to split the region into 2 sections because of the 0 < y, but I am not sure. Thanks for any help.
Ok. I found C to be 2/19. I integrated over the 2 regions. For the smaller region I got 3/2C, and for the larger region I got 8C. So 3/2C + 8C = 1, therefore C = 1/9.5 or 2/19.
To find the covariance would I use the formula E(XY) - E(X)E(Y)? To find E(XY) would I integrate again over the regions but this time have xy inside the integral? Would the C be 2/19 for both sides or would I need to split it up for both sides?
As has already been suggested, since the joint pdf is constant you can use simple geometry:
You require c(area of trapezium - area of triangle) = 1.
Area of trapezium = 5(1 + 6)/2 = 35/2.
Area of triangle = (4)(4)/2 = 8.
c(35/2 - 8) = 1 => c = 2/19 so yes you're correct.
Yes. Yes. Yes it would - c = 2/19 is the pdf so why on Earth would you split it up??