Question

Given that `barx` is the mean and `sigma^(2)` is the variance of `n` observations `x_(1),x_(2),………….x_(n)`. Prove that the mean and variance of the observations `ax_(1),ax_(2),ax_(3),………….ax_(n)` are `abarx` and `a^(2)sigma^(2)` , respectively, `(a!=0)`

Answer 1

Mean of `n` observations
`barx=(x_(1)+x_(2)+…………..+x_(n))/n=(sumx_(i))/nimpliessumx_(i)=n.barx`………….1
Variance `sigma^(2)=(sum(x_(i)-barx)^(2))/n`
`implies sum(x_(i)-barx)^(2)= nsigma^(2)`…………….2
Now mean of observation `ax_(1),ax_(2),………..,ax_(n)`
`barx=(ax_(1)+ax_(2)+..............+ax_(n))/n=(a(x_(1)+x_(2)+..........+x_(n)))/n`
`=(asumx_(i))/n=(a.nbarx)/n=abarx`............3
Hence Proved.
and variance `=((sumX_(i)-barX)^(2))/n=1/nsum(ax_(i)-abarx)^(2)`
`=a^(2)sigma^(2)` Hence Proved.