InterviewSolution
Saved Bookmarks
InterviewSolution
| 1. |
Find the mean, standard deviation and variance of first n natural numbers. |
|
Answer» First n natural number are 1,2,3,n. `"Mean, "barx=((1+2+3+...+n))/(n)=(1)/(n)cdot(1)/(2)n(n+1)=(1)/(2)(n+1)" "[because (1+2+3+...+n)=(1)/(2)n(n+1)]` `therefore" variance," sigma^(2)=(Sigmax_(i)^(2))/(n)-barx^(2)` `=(Sigman^(2))/(n)-{(1)/(2)(n+1)}^(2)` `=(1)/(n)cdot(n(n+1)(2n+1))/(6)-(1)/(4)(n+1)^(2)" "[because Sigman^(2)=(1)/(6)n(n+1)(2n+1)]` `={((b+1)(2n+1))/(6)-((n+1)^(2))/(4)}` `=(n+1)cdot{((2n+1))/(6)-((n+1))/(4)}` `=((n+1)(n-1))/(12)=((n^(2)-1))/(12).` `therefore" variance,"sigma^(2)=((n^(2)-1))/(12).` Standard deviation, `sigma=sqrt((n^(2)-1)/(12))=(1)/(2).sqrt((n^(2)-1)/(3)).` |
|