Question

QUESTION 4 Suppose Xis a random variable with probability density function f(x) and Y is a random variable with density function f,(x). Then X and Y are called independent random variables if their joint density function is the product of their individual density functions: x, y We modelled waiting times by using exponential density functions if t <0 where μ is the average waiting time. In the next example we consider a situation with two independent waiting times. The joint density function of X and Y is a function f of two variables such that the probability (xy) lied in the region Dis

Answer 1

"The Given information rightarrow X is a random variable with probability density function f_1(x) Y is a "" "" f_2 (x) rightarrow f (x, y) = f_1 (x) f_2(y) f (t) = [0 if t < 0 m^-1 e^-t/m if t greaterthanorequalto 0 implies f (x) = {0 x < 0 1/2 e^-x/2 x greaterthanorequalto 0 m = 2 f (y) = {0 y < 0 1/20 e^-y/20 y greaterthanorequalto 0 m = 20 rightarrow P (X + Y < 30) = integral^30_0 f (x) (integral^30 - x_0 f (y)dy)dx = 1/20 1/2 integral^30_0 e^-x/2 (integral^30 - x_0 e^-y/20 dy) dx = 20/40 integral^30_0 e^-x/2 (1 - e6-3/20 -3/2)dx = 1 - 1/9 e^-15 - 10/9 e^-1.5 = 0.7521"