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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. - Mathematics

Sum

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.

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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`

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