Question

a pair of random variables X and Y takes values between 0 and 1 and has P (X<=x, Y<= y)=x^3y^2 when 0<=x, y<=1. Find the density function.

Answer 1

The definition of the density function f(x,y)
\[F(x_0,y_0)=\int\limits_{-\infty}^{y_0}\int\limits_{-\infty}^{x_0} f(x,y)\;\mathrm{d}x\;\mathrm{d}y\] In this case it's
\[F(x_0,y_0)=x_0^3y_0^2=\int\limits_0^{y_0}\int\limits_0^{x_0} f(x,y)\;\mathrm{d}x\;\mathrm{d}y.\] It is not hard to see that
\[f(x,y)=6x^2y.\]