It is known that the joint probability density function of two-dimensional random variables (x, y) is f (x, y) = 2E ^ (- 2x-y), x > 0, Y > 0; f (x, y) = 0 (others) -------------------------------------------------------------------------------------------------- It is known that the joint probability density function of two-dimensional random variables (x, y) is F (x, y) = 2E ^ (- 2x-y), x > 0, Y > 0; f (x, y) = 0 (others) Find the probability of (x, y) falling in the region D: x > = 0, Y > = 0 and X + Y > = 1

It is known that the joint probability density function of two-dimensional random variables (x, y) is f (x, y) = 2E ^ (- 2x-y), x > 0, Y > 0; f (x, y) = 0 (others) -------------------------------------------------------------------------------------------------- It is known that the joint probability density function of two-dimensional random variables (x, y) is F (x, y) = 2E ^ (- 2x-y), x > 0, Y > 0; f (x, y) = 0 (others) Find the probability of (x, y) falling in the region D: x > = 0, Y > = 0 and X + Y > = 1

P=∫(0-->1)e^(-y)dy∫(0-->1-y)2e^(-2x)dx
=∫(0-->1)e^(-y)(1-e^(2(1-y))dy
=∫(0-->1)(e^(-y)-e^2e^y)dy=(1-e)(1+e^2)