two balanced dice are rolled write down the sample space plo

two balanced dice are rolled write down the sample space plotting the outcomes as coordinates in a cartesian coordinate system i.e xy plane . Let x be the largest digit facing up. (b) use the sample space to obtain the probability mass function of x.(c) find the expected value E[X] variance V[X] and the deviation

Solution

Sanple space for 2 dice rolling together= (1,1),(1,2),(1,3),(1,4),(1,5),(1,6), (2,1),(2,2),(2,3),(2,4),(2,5),(2,6), (3,1),(3,2),(3,3),(3,4),(3,5),(3,6), (4,1),(4,2),(4,3),(4,4),(4,5),(4,6), (5,1),(5,2),(5,3),(5,4),(5,5),(5,6), (6,1),(6,2),(6,3),(6,4),(6,5),(6,6).
x= largest digit facing up .....It is the maximum no. of the outcomes of the 2 dice.

now,
x = 1when maxm of both no.s =1
  = 2 when maxm of both no.s =2
  = 3 when maxm of both no.s =3
  = 4 when maxm of both no.s =4
  = 5 when maxm of both no.s =5
= 6 when maxm of both no.s =6

so,this is the pmf of x.
p(x)= 1/36 when x=1
     = 3/36 when x=2
= 5/36 when x=3
= 7/36 when x=4
= 9/36 when x=5
     = 11/36 when x=6

E(x)= sum( x*p(x) ) = 1*(1/36) + 2* ( 3/36) + 3* ( 5/36) + 4* (7 /36) + 5*(9/36) + 6*(11/36) =4.4722
V(x)= E(x^2) - (E(x))^2 = 21.9722-20.00077 =1.9715
standard deviation=1.404083551