There are 4 cards numbered 1, 3, 5 and 7, one number on one card. Two cards are drawn at random without replacement. Let X denote the sum of the numbers on the two drawn cards. Find the mean 'and variance of X.
Solution
X can be 4, 6, 8, 10, 12
n(S) = 12
P(X = 4) = {1, 3} or {3, 1}
P(X =4) = `2/12`
P (X = 6) = {1,5} or {5,1}
P(X = 6 ) = `2/12`
P(X = 8) = {1,7} or {7,1} or {3 , 5} or { 5, 3}
P(X = 8) = `4/12`
P(X = 10 ) = {3, 7} or {7, 3 }
P(X = 10 ) = `2/12`
P(X =12) = {5, 7} or { 7, 5}
P(X = 12) = `2/12`
Mean = `sum_iP_iX_i = 4 (2/12 )+6(2/12) +8(4/12) +10(2/12) +12(2/12) =1/12(8+12+32+20+24)`
`=96/12 = 8`
`sum_i P_i X_i ^2 = 16 (2/12) + 36(2/12) + 64 (4/12) + 100(2/12) + 144(2/12)`
`=1/12 (32 +72 +256 +200 +288 ) = 848/12`
Varience `= 848/12 -8^2 = 70 .666 - 64 = 6. 666`
