Question

Let X be the time of the next earthquake in San Francisco and Y the time of the next earthquake in Los Angeles. X has exponential density λe^−λx, x > 0 and Y has exponential density 2λe^−2λy, y > 0 (λ > 0). Find the probability that the next earthquake occurs in Los Angeles.

A. 1/3

B. 1/2

C. 2/3

D. 1/8

Answer 1

Answer:

Given,

fx,y(x,y) = fx(x)*fy(y)

Here x & y are independent

= 2^2*e^-(x+2y) , x>0 & y>0

P(Y < X) = fx,y(x,y) dx dy

= 2^2 e^-x*e^-2y dx dy

= 2^2 e^-2y*[e^-x/-] dy

= 2 [ e^-3y / (-3)]

= 2/3

Option C