Question

Find the mean and variance for the first n natural numbers.

Answer 1

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`