Aspire Faculty ID #13955 · Topic: JAMIA MILLIA ISLAMIA MCA 2020 · Just now
JAMIA MILLIA ISLAMIA MCA 2020

What will be the mean and variance for the first $n$ natural numbers?

Solution

For $1,2,\dots,n$: Mean $= \dfrac{1+\cdots+n}{n} = \dfrac{n(n+1)/2}{n} = \dfrac{n+1}{2}$. Variance $= \dfrac{1}{n}\sum_{k=1}^{n}\left(k-\dfrac{n+1}{2}\right)^{2} = \dfrac{n^{2}-1}{12}$. $\boxed{\dfrac{n+1}{2},\ \dfrac{n^{2}-1}{12}}$
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