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Bernard takes successive readings of an instrument, but the variance of his readings increases linearly with each observation, so that $\sigma_r^2=A+Br$. Find the variance of the mean of $n$ successive readings he takes.

I tried to write out successive sigma r and thought about the variance being $E(X^2)-(E(X))^2$ but to no avail.

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up vote 0 down vote accepted

Hint: Let $X_1, X_2, \dots, X_n$ be the $n$ successive readings, with variances $A + B, A+2B, \dots, A+nB$ respectively. Now, use the fact that $V(X+Y) = V(X) + V(Y)$ (assuming that readings are independent) to calculate the variance of $X_1 + X_2 + \dots + X_n$. From there you can easily calculate the variance of the mean.

I'm not going to show you all the work, because I assume this is homework, but I think that should be plenty enough of a hint.

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$Var(X_1+X_2+...+X_n)=nA+B(1+2+...+n)=nA+\frac{1}{2}Bn(n+1)$ Do I divide by n to get the variance of the mean? so $A+\frac{1}{2}B(n+1)$ – bbr4in Jan 23 '13 at 22:54
@user52187 $V(X/n) = \frac{1}{n^2}V(X)$. – Jonathan Christensen Jan 23 '13 at 23:04

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