Answer :

Given : There are 5 cards, numbers 1 to 5, one number on each card. Two cards are drawn at random without replacement. Let X denote the sum of the numbers on the two cards drawn.


To find : mean (𝓊) and variance (σ2) of X


Formula used :



Mean = E(X) =


Variance = E(X2) -


There are 5 cards, numbers 1 to 5, one number on each card. Two cards are drawn at random without replacement.


X denote the sum of the numbers on two cards drawn


The minimum value of X will be 3 as the two cards drawn are 1 and 2


The maximum value of X will be 9 as the two cards drawn are 4 and 5


For X = 3 the two cards can be (1,2) and (2,1)


For X = 4 the two cards can be (1,3) and (3,1)


For X = 5 the two cards can be (1,4) , (4,1) , (2,3) and (3,2)


For X = 6 the two cards can be (1,5) , (5,1) , (2,4) and (4,2)


For X = 7 the two cards can be (3,4) , (4,3) , (2,5) and (5,2)


For X = 8 the two cards can be (5,3) and (3,5)


For X = 9 the two cards can be (4,5) and (4,5)


Total outcomes = 20


P(3) = =


P(4) = =


P(5) = =


P(6) = =


P(7) = =


P(8) = =


P(9) = =


The probability distribution table is as follows,



Mean = E(X) = = x1P(x1) + x2P(x2) + x3P(x3) + x4P(x4) + x5P(x5) + x6P(x6) + x7P(x7)


Mean = E(X) = 3() + 4() + 5() + 6() + 7() + 8() + 9()


Mean = E(X) = + + + + + + = = = 6


Mean = E(X) = 6


= = 36


E(X2) = = P(x1) + P(x2) + P(x3) + P(x4) + P(x5) + P(x6) + P(x7)


E(X2) = () + () + () + () + () + () + ()


E(X2) = + + + + + + = = = 39


E(X2) = 39


Variance = E(X2) - = 39 – 36 = 3


Variance = E(X2) - = 3


Mean = 6


Variance = 3


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