I have a question about joint CDFs. My understanding was given a joint PDF, the joint CDF was the integral of the joint PDF from -inf to +inf for all the random variables defined. This joint CDF should be equal to 1. However, in the question below I see a contradiction. When I integrate, the joint CDF equals 1. However, calculating the area under the curve just looking at the bounds gives me an answer of 1/4. I am not sure why I am getting different answers. Additionally, if the area under the curve is equal to 1/4 does this make the joint PDF I was given not a valid PDF?
Joint PDF problem described above:
You just missed the factor of $4$ in the second calculation. The area of the region is not the value of the integral of $f_{x,y}$ since the value of $f_{x,y}$ is $4$ in the region. You have to multiply the area by $4$.