If the mean and variance of the observations x_(1), x_(2), x_(3),…,x_(n) are barx and sigma^(2) respectively and a be a nonzero real number, then show that the mean and variance of ax_(1),ax_(2),ax_(3),…,ax_(n)" are "abarx and a^(2) sigma^(2) respectively.

If the mean and variance of the observations x_(1), x_(2), x_(3),…,x

1.	If the mean and variance of the observations x_(1), x_(2), x_(3),…,x_(n) are barx and sigma^(2) respectively and a be a nonzero real number, then show that the mean and variance of ax_(1),ax_(2),ax_(3),…,ax_(n)" are "abarx and a^(2) sigma^(2) respectively.
Answer» Solution :Let `barx` be the mean of `x_(1),x_(2),x_(3),x_(n)` and a be nonzero real number. Then, `barx=(1)/(n)(x_(1)+x_(2)+x_(3)+ +x_(n)).` Let `y_(i)=ax_(i)" for each i=1, 2, 3,n. Then",` `bary=(1)/(n)(y_(1)+y_(2)+y_(3)+...+y_(n))` `=(1)/(n)(ax_(1)+ax_(2)+ax_(3)+ +ax_(n))=acdot(1)/(n)(x_(1)+x_(2)+x_(3)+...+x_(n))=abarx.` Thus, `bary=a barx.` Now, the variance of new observations is given by `"variance "(y)=sigma_(1)^(2)` `=(1)/(n)cdotoverset(n)underset(i=1)Sigma(y_(i)-bary)^(2)` `=(1)/(n)CDOT overset(n)underset(i=1)Sigma(ax_(i)-abarx)^(2)" "[because y_(i)=ax_(i)" for each i and "bary=abarx]` `=a^(2)cdot(1)/(n)cdotoverset(n)underset(i=1)Sigma(x_(i)-barx)^(2)` `=a^(2)cdot{"variance"(x)}=a^(2)sigma^(2).` `THEREFORE" new variance"=a^(2)sigma^(2).` `"REMARK "sigma_(1)=SQRT(a^(2)sigma^(2))=\|a\|cdot sigma.`