Question

Let X and Y be independent exponential random variables with pa-
rameter ? = 1. Given that X and Y are independent, their joint pdf is given by the product of the
individual pdfs of X and Y , that is, f_X,Y(x, y) = f_X(x) f_Y(y).
The joint pdf is defined over the same set of x-values and y-values that
the individual pdfs were defined for. Using this information, calculate
P (X ? Y ? 2) where you can assume t is a positive number.

Answer 1

"Note: In given problem small mistake is there i.e p(x - y lessthanorequalto t) but it is given as p(x - y lessthanorequalto 2) so p(x - y lessthanorequalto t) is right. Since X and Y be independent exponential random variable with parameter (lambda = 1) f(x) = 1/lambda e^- x/lambda = 1/1 e^- x/1 = e^- x f(y) = e^- y f(x, y) = f(x).f(y) f(x, y) = e^-(x + y) p(x - y lessthanorequalto t) = ? Let the transformation, (x - y) = u Rightarrow x = u + v(y = v) Jacobean