I roll a fair die four times Let X be the number of differen

I roll a fair die four times. Let X be the number of different outcomes that I saw. (For example, if the die rolls are 5, 3, 6, 6 then X = 3 because the different outcomes are 3, 5 and 6.) Find the mean and variance of X.

Solution

X = no. of different outcomes in rolling a fair die 4 times....

so,X can take values 1 ,2 ,3 and 4 only!

X = 1 with probability 6 /1296 ( all outcomes are same = 6 )
  = 2 with probability  210 /1296 ( 2 outcomes are distinct = ( 6*5*2*2)+(6*1*1*5)+(6*1*5*2) )

= 3 with prob. 720 / 1296 ( 3 outcomes are distinct = (6*5*4*3)+(6*5*2*4)+(6*1*5*4) )
= 4 with prob. 360 / 1296 ( 4 distinct outcomes = ( 6*5*4*3) )
mean = 3.106481481.....

variance = 0.44699503...