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Question
Find the mean and variance of the number randomly selected from 1 to 15
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Solution
The sample space of the experiment is S = {1, 2, 3,…,15}
Let X denotes the selected number.
Then X is a random variable which can take values 1, 2, 3, …, 15.
∴ P(1) = P(2) = P(3) = … = P(15) = `1/15`
E(X) = `sum_("i" = 1)^"n" x_"i""P"_"i"`
= `1 xx 1/15 + 2 xx 1/15 + 3 xx 1/15 + ... + 15 xx 1/15`
= `(1 + 2 + 3 + ... + 15) xx 1/15`
= `((15 xx 16)/2) xx 1/15`
= 8
Var(X) = `(sum_("i" = 1)^"n"x_"i"^2"p"_"i") - (sum_("i" = 1)^"n"x_"i""p"_"i")^2`
= `1^2 xx 1/15 + 2^2 xx 1/15 + 3^2 xx 1/15 + ... + 15^2 xx 1/15 - (8)^2`
= `(1^2 + 2^2 + 3^2 + ... + 15^2) xx 1/15 - (8)^2`
= `((15 xx 16 xx 31)/6) xx 1/15 - (8)^2`
= 82.67 – 64
= 18.67
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