Question

Plz help me to do both quetions.

6. The continuous random variables X and Y have joint density 2e2)for and y 20 otherwise Find P(X >Y). (Answer: 3) 7. The continuous random variables X and Y have joint probability density function 10y for 01 and 0 yr fzy(z,y)=ï o otherwise Find the marginal density function fy(y). Show your work. Do not forget to indicate where the density is non-zero.

Answer 1

6)

P(X>Y)=$\int_{0}^{\infty }$$\int_{0}^{x }$ f(x,y) dy dx =$\int_{0}^{\infty }$$\int_{0}^{x }$ 2e^-xe^-2y dy dx =$\int_{0}^{\infty }$ 2e^-x(-e^-2y/2)|^x₀ dx

=$\int_{0}^{\infty }$ e^-x(1-e^-2x) dx =$\int_{0}^{\infty }$(e^-x -e^-3x) dx =(-e^-x+e^-3x/3)|^$\infty$₀ =(1-1/3) =2/3

7)

f_y(y) =$\int_{y}^{1}$ f(x,y) dx =$\int_{y}^{1}$ 10x²y dx = 10x³y/3 |¹_y =(10/3)*y*(1-y³) for 0<=y<=1