Talk:StudentDeduction

For mean value zero:

Case of 2 measurements

\[ \mathrm{Nor}(x)=\frac{1}{\sqrt{2\pi}} \exp(-x^2/2) \]

\[ R_2= \int_{-\infty}^{\infty} \mathrm d x_1 \int_{-\infty}^{\infty} \mathrm d x_2 \ \mathrm{Nor}(x_1)\ \mathrm{Nor}(x_2)\ \frac{\big((x_1 + x_2)/2\big)^2 } {(x_1-x_2)^2} \]

Case of 3 measurements

\[ \mathrm{Nor}(x)=\frac{1}{\sqrt{2\pi}} \exp(-x^2/2) \]

\[ M=M(x_1,x_2,x_3))=\frac{x_1+x_2+x_3}{3} \]

\[ s^2=\frac{ (x_1-M)^2+ (x_2-M)^2+ (x_3-M)^2 }{6} \]

\[ R_3= \int_{-\infty}^{\infty} \mathrm d x_1 \int_{-\infty}^{\infty} \mathrm d x_2 \int_{-\infty}^{\infty} \mathrm d x_3 \ \mathrm{Nor}(x_1)\ \mathrm{Nor}(x_2)\ \mathrm{Nor}(x_3)\ \frac{M^2 } {s^2} \]

Case of 4 measurements

\[ \mathrm{Nor}(x)=\frac{1}{\sqrt{2\pi}} \exp(-x^2/2) \]

\[ M=\frac{x_1+x_2+x_3+x_4}{4} \]

\[ s^2=\frac{ (x_1-M)^2+ (x_2-M)^2+ (x_3-M)^2+ (x_4-M)^2 }{12} \]

\[ R_4= \int_{-\infty}^{\infty} \mathrm d x_1 \int_{-\infty}^{\infty} \mathrm d x_2 \int_{-\infty}^{\infty} \mathrm d x_3 \int_{-\infty}^{\infty} \mathrm d x_4\ \mathrm{Nor}(x_1)\ \mathrm{Nor}(x_2)\ \mathrm{Nor}(x_3)\ \mathrm{Nor}(x_4)\ \frac{M^2} {s^2} \]