5.2–3

5.2–3 Use indicator random variables to compute the expected value of the sum of n dice. Let $X1,X2,X3,X4,X5,X6$ be the indicator variable that capture the no of 1,2,3,4,5,6 on a dice. Let X the indicator variable that denotes the sum when we throw the dice n times or throw n dice. Expected value is the Mean of the random variable so $$E[X] = 1 E[X1]+ 2 E[X2]+3E[X3]+4E[X4]+5 E[X5]+6 E[X6]$$ the above equation can be written as $$E[X] = \sum_{i = 1}^6 i E[Xi] $$ the no of 1 side in n dice throws $E[X1] = \sum_{i = 1}^n 1/6 $ $$E[X1] = 1/6(1+1+1....1) $$ sum of n 1's is n $$E[X1] = n/6 $$ $$E[X] = \sum_{i = 1}^6 i E[Xi] $$ $$E[X] = \sum_{i = 1}^6 i n/6 $$ $$E[X] = n/6 \sum_{i = 1}^6 i $$ $$E[X] = n/6 (1+2+3+4+5+6) $$ $$E[X] = 21n/6 $$