Science Yoshio Aug 10th, 2023 at 8:00 PM 8 0
Probability And Statistics
機率與統計Probability
Topics include basic combinatorics, random variables, probability distributions, Bayesian inference, hypothesis testing, confidence intervals, and linear regression. 條件機率(conditionally probability). A, B 為事件. · 獨立事件 (independent events). 機率空間（2）機率的意義（Probability space-2. The meanings of probability）
Probability Density Functions
\(Let\;\int_0^\infty e^{-x^2}dx\;equal\;to\;I\)
\(I^2\;=\;\int_0^\infty e^{-x^2}dx\int_0^\infty e^{-y^2}dy\)
\(=\;\int_0^\infty\int_0^\infty e^{-(x^2+y^2)}dxdy\)
\(=\;\int_0^\frac{\mathrm\pi}2\int_0^\infty e^{-r^2}r\;dr\;d\theta\)
\(=\;\int_0^\frac{\mathrm\pi}2d\theta\left.\left[-\frac12e^{-r^2}\right]\right|_0^\infty\)
\(=\;\frac{\mathrm\pi}2(0+\frac12)\;=\;\frac{\mathrm\pi}4\;\)
\(I\;=\;\frac{\sqrt{\mathrm\pi}}2\;\)
Proof:
\(\int_{-\infty}^\infty\frac1{\sqrt{2\mathrm\pi}\sigma}e^{-\frac12{(\frac{x-\mu}\sigma)}^2}dx\;=\;1\)
\(Let\;z\;=\;\frac{x-\mu}\sigma,\;\;\;dz\;=\;\frac1\sigma dx\)
\(=\int_{-\infty}^\infty\frac1{\sqrt{2\mathrm\pi}}e^{-\frac12z^2}dz\)
\(=\int_0^\infty\frac2{\sqrt{2\mathrm\pi}}e^{-\frac12z^2}dz\)
\(let\;z\;=\;\sqrt2y,\;dz\;=\sqrt2dy,\;z^2\;=\;2y^2\)
\(=\int_0^\infty\frac{2\sqrt2}{\sqrt{2\mathrm\pi}}e^{-y^2}dy\)
\(=\frac{2\sqrt2}{\sqrt{2\mathrm\pi}}\cdot\frac{\sqrt{\mathrm\pi}}2\)
\(=1\)
\(x\sim N(\mu,\;\sigma^2)\)
動差生成函數 Moment Generating Function
\( \begin{align*} {M_X}\left( t \right) &= E\left[ {{e^{tX}}} \right] \hfill \\ &= \int_{ - \infty }^\infty {{e^{tx}}{f_X}(x)} dx \hfill \\ &= \int_{ - \infty }^\infty {{e^{tx}}\frac{1}{{\sqrt {2\pi } \sigma }}{e^{ - \frac{{{{\left( {x - \mu } \right)}^2}}}{{2{\sigma ^2}}}}}} dx \hfill \\ &= \frac{1}{{\sqrt {2\pi } \sigma }}\int_{ - \infty }^\infty {{e^{\frac{{2{\sigma ^2}tx - {{\left( {x - \mu } \right)}^2}}}{{2{\sigma ^2}}}}}} dx \hfill \\ &= \frac{1}{{\sqrt {2\pi } \sigma }}{e^{\frac{{ - {\mu ^2}}}{{2{\sigma ^2}}}}}\int_{ - \infty }^\infty {{e^{\frac{{ - \left( {{x^2} - 2\left( {{\sigma ^2}t + \mu } \right)x + {{\left( {{\sigma ^2}t + \mu } \right)}^2}} \right)}}{{2{\sigma ^2}}}}}{e^{\frac{{{{\left( {{\sigma ^2}t + \mu } \right)}^2}}}{{2{\sigma ^2}}}}}} dx \hfill \\ &= \frac{1}{{\sqrt {2\pi } \sigma }}{e^{\frac{{ - {\mu ^2}}}{{2{\sigma ^2}}}}}{e^{\frac{{{{\left( {{\sigma ^2}t + \mu } \right)}^2}}}{{2{\sigma ^2}}}}}\int_{ - \infty }^\infty {{e^{\frac{{ - {{\left( {x - \left( {{\sigma ^2}t + \mu } \right)} \right)}^2}}}{{2{\sigma ^2}}}}}} dx \hfill \\ &= {e^{\frac{{{\sigma ^2}{t^2} + 2\mu t}}{2}}}\underbrace {\frac{1}{{\sqrt {2\pi } \sigma }}\int_{ - \infty }^\infty {{e^{\frac{{ - {{\left( {x - \left( {{\sigma ^2}t + \mu } \right)} \right)}^2}}}{{2{\sigma ^2}}}}}} dx}_{ = 1{\text{ }}\left( {{\text{pdf of normal }}\mathcal{N}\left( {\mu+\sigma^2 t ,{\sigma ^2}} \right)} \right)} \hfill \\ &= {e^{\mu t + \frac{{{\sigma ^2}{t^2}}}{2}}} \hfill \\ \end{align*} \)
