3 years ago
Suppose X and Y are random variables with joint density function f(x, y)=(0.1 e^(-(0.5x+0.2y)) if >=0, y >=0 , 0 otherwise. Find the following probabilities. P(Y >=1)
1 Answers
Best Answer
Gaby70 Staff answered 3 years ago
Gaby70 Staff answered 3 years ago
Since f(x,y) is non-zero only when both x, y are non-negative, we can write
P(Y >=1)= int_(- infty)^infty int_(1)^infty f(x,y)dydx= int_0^infty int_1^infty f(x,y)dydx
= int_0^infty int_1^infty0.1e^(-(0.5x+0.2y))dydx
Use the property n^(a* b)=n^a* n^b
=0.1 int_0^infty int_1^infty e^(-0.5x)* e^(-0.2y)dydx
=0.1[ int_0^infty e^(-0.5x)dx][ int_1^infty e^(-0.2y)dy]
=0.1[- (1)/(0.5)e^(-0.5x)]_0^infty[- (1)/(0.2)e^(-0.2y)]_1^infty
=0.1[ (1)/(0.5)][ (1)/(0.2)e^(-0.2)]
=0.1[2][5e^(-0.2)]=e^(-0.2) ~ 0.8187