Question

Suppose that the standard normal random variables X and Y are independent. Find P(0 < X<Y). 8 O 1 4T 0 1 8л Ala

Answer 1

Since X and Y are independnet. So thier joint pdf is given by

$f(x,y)={1\over \sqrt{2\pi}}e^{-{1\over 2}x^2} \times {1\over \sqrt{2\pi}}e^{-{1\over 2}y^2} \\ \\ ~~~~~~~~~~~~~~={1\over 2\pi }e^{-{1\over 2}(x^2+y^2)}~~~~~~~~~,~~~~~-\infty<x, y< \infty$

Hence, the required probability is

$P(0<X<Y)=\int_{0}^{\infty}\int_{0}^{y} f(x,y)dxdy \\ ~~~~~~~~~~~~~~~~~~~~~~~~={1\over 2\pi } \int_{0}^{\infty}\int_{0}^{y} e^{-{1\over 2}(x^2+y^2)}dxdy \\ ~~~~~~~~~~~~~~~~~~~~~~~~={1\over 2\pi } \left ( {\pi \over 4} \right ) \\ ~~~~~~~~~~~~~~~~~~~~~~~~={1\over 8 }$

Thus, correct Option is 1st