StudentDeduction

StudentDeduction is set of formulas used to deduce the Student Distribution as distribution of the ratio (sample mean)/(sample spread).

Deduction

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

\[ \mathrm{Chi}_k(x)=\frac{x^{k/2-1}\ \exp(-x/2)}{2^{k/2}\ \Gamma(k/2)} \]

\[ \varphi(x)=\sqrt{N}\ \mathrm{Nor}\!\left(\sqrt{N} x\right) \]

\[ \rho(u) = \int_0^\infty \mathrm d x \int_{-\infty}^\infty \mathrm d y \ \varphi(y) \ \mathrm{Chi}_{N-1}(x)\ \delta\!\left( \frac{y}{\sqrt{x}} - u \right) \]

\[ \rho(u) = \int_0^\infty \mathrm d x \int_{-\infty}^\infty \mathrm d y \ \varphi(y) \ \mathrm{Chi}_{N-1}(x)\ \sqrt{x}\ \delta\!\left( y - \sqrt{x}\ u \right) \]

\[ \rho(u) = \int_0^\infty \mathrm d x\ \varphi\!\left(\sqrt{x}\ u\right) \ \mathrm{Chi}_{N-1}(x)\ \sqrt{x} \]

\[ \rho(u) = \int_0^\infty \mathrm d x\ \sqrt{N}\ \mathrm{Nor}\!\left(\sqrt{N}\sqrt{x}\ u\right)\ \mathrm{Chi}_{N-1}(x)\ %f(x) \ \sqrt{x} \]

\[ \rho(u) = \int_0^\infty \mathrm d x\ \frac{\sqrt{N}}{\sqrt{2\pi}} \ \exp\!\left( - \left(\sqrt{N}\sqrt{x}\ u\right)^2\!/\ 2\right) \mathrm{Chi}_{N-1}(x)\ \sqrt{x} \]

\[ \rho(u) = \int_0^\infty \mathrm d x\ \frac{\sqrt{N}}{\sqrt{2\pi}} \ \exp\!\left( - \frac{N u^2}{2}\ x\right) \frac{x^{(N-1)/2-1}\ \exp\left(-\frac{x}{2}\right)} { 2^{(N-1)/2}\ \Gamma \left( \frac{N-1}{2} \right) } \sqrt{x} \]

\[ \rho(u) = \frac{\sqrt{N}}{\sqrt{\pi}\ 2^{N/2}\ \Gamma \left( \frac{N-1}{2} \right)} \ \int_0^\infty \mathrm d x\ \exp\!\left( - \frac{N u^2}{2}\ x -\frac{x}{2}\right) x^{N/2-1} \]

\[ \rho(u) = \frac{\sqrt{N}}{\sqrt{\pi}\ \Gamma \left( \frac{N-1}{2} \right)} \ \int_0^\infty \mathrm d x\ \exp\!\left( - (N u^2 \!+\!1) \ x\right) x^{N/2-1} \]

\[ \rho(u) = \frac{\sqrt{N}}{\sqrt{\pi}\ \Gamma \left( \frac{N-1}{2} \right)} \ \int_0^\infty \mathrm d x\ \exp(-x)\ x^{N/2-1} \ \frac{1}{(N u^2 \!+\!1)^{N/2}} \]

\[ \rho(u) = \frac{\sqrt{N}}{\sqrt{\pi}\ \Gamma \left( \frac{N-1}{2} \right)} \ \Gamma\! \left( \frac{N}{2} \right) \frac{1}{(N u^2 \!+\!1)^{N/2}} \]

\[ \frac{1}{\sqrt{N(N\!-\!1)}}\ \rho\!\left(\!\frac{t}{\sqrt{N(N\!-\!1)}}\!\right) = \frac{1/\sqrt{N\!-\!1}}{\sqrt{\pi}\ \Gamma \left( \frac{N-1}{2} \right)} \ %\int_0^\infty \mathrm d x\ \exp(-x)\ x^{N/2-1} \ \Gamma\! \left( \frac{N}{2} \right) \frac{1}{\left(\frac{t^2}{N-1} \!+\!1\right)^{N/2}} \]

\[u=\frac{t}{\sqrt{N(N\!-\!1)}}\]

\[ t=u\ \sqrt{N(N\!-\!1)} \]

\[\mathrm{Student}_{N-1}(t)= \frac{\Gamma\! \left( \frac{N}{2} \right)}{\sqrt{\pi\ (N\!-\!1)}\ \Gamma \left( \frac{N-1}{2} \right)} \ \frac{1}{\left(\frac{t^2}{N-1} \!+\!1\right)^{N/2}} \]