Question

A dice is rolled once. The random variable X is defined to be 1 if the outcome is 1 or 3,
to be 2 if the outcome is 2 or 4, and to be 3 if the outcomes is 5 or 6.

a) Find f_x(x) and F_x(x)

b) Find f_x(x/M) and F_x(x/M)  if the event M is outcome to be even, M = {2,4,6}.

Answer 1

a)below is pmf of X

cause f(x=1)=2/6 =1/3 (As 1 or 3 are two outcomes out of 6)

similalry f(x=2)=1/3

f(x=3)=1/3

here f(x)=1/3 where x=1,2,3

from above CDF F(x) =x/3 where x =1,2,3  

b)f(M) =1/2 (As these are 3 events out of 6)

also f(x)=1/3

f(x and M) =1/6 (as here for each value of x ; we have one common event out of 6)

here as f(x and M)=f(x)*f(M) ; therefore x and M are independent.

hence f(x|M) =f(x)= 1/3 where x=1,2,3

and F(x|M) =F(x)=x/3 where x=1,2,3