GATE 2012 MA

Question

ANSWER GIVEN 0.73-0.77

Answer 1

The first thing to understand here is :

max(X,Y) < 2  implies that  X < 2 and Y < 2..

So P(max(X,Y) < 2) means P((X < 2)  and  (Y < 2))   

Now given X and Y are independent random variables..

Hence P(X < 2  and Y < 2)  =  P(X < 2) . P(Y < 2)

Now P(X < 2)   =   ∫^∞_-∞  f(x) dx where f(x) is probability density function..

                 =   ∫⁰_-∞  f(x) dx  + ∫²₀ f(x) dx

                 =  0  +  e^-x / (-1)  |²₀

                 =  1  -  1/e²

                 =  0.865  [ correct to 2 decimal places ]

Similarly P(Y < 2)   =  0.865

Hence , P(max(X,Y) < 2)  =   P(X < 2  and Y < 2)  

                                      =  P(X < 2) . P(Y < 2)

                                      =  0.865 * 0.865

                                      =  0.75  [ correct to 2 decimal places ]