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Question
Find the mean and variance for the first n natural numbers.
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Solution
First n natural numbers: 1, 2, 3, ….., n
Mean `overline x = (1 + 2 + 3 + .... + n)/n = 1/n. (n(n +1))/2`
= `(n + 1)/2` ....`["Sum of the first natural numbers" (n(n +1))/2]`
`sum x_i^2 = 1^2 + 2^2 + 3^2 + .... + n^2`
= `(n(n + 1)(2n + 1))/6`
Variance = `(sum(x_i - overlinex)^2)/n = 1/n^2 [n sum x_i^2 - (sumx_i)^2]`
= `1/n^2 [n(n(n+ 1)(n + 2))/6 - (n^2(n + 1)^2)/4]`
= `1/12 [2(n+ 1)(2n+ 1) - 3(n+ 1)^2]`
= `(n+ 1)/12 [2(2n + 1) - 3(n+ 1)]`
= `(n+ 1)/12 [4n + 2 - 3n - 3]`
= `((n+ 1)(n- 1))/12`
= `(n^2 - 1)/12`
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