7: Probability and Statistics
- Page ID
- 121980
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)
( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\id}{\mathrm{id}}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\kernel}{\mathrm{null}\,}\)
\( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\)
\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\)
\( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)
\( \newcommand{\vectorA}[1]{\vec{#1}} % arrow\)
\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow\)
\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vectorC}[1]{\textbf{#1}} \)
\( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)
\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)
\( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)To further improve communication of experimental results, the combination of probability (expected outcomes) with statistics (methodology to reduce data) are covered.
- Understand the meaning of central moment of probability density functions through calculation
- Convert finite quantity of data to have indication of true value of observation
A. Introduction of Terminology
- Probability – likelihood of particular event occurrence
- Statistics – post processing analysis tool for interpretation of probability
- Samples – discrete data points of collection in statistics that indicate probability
- Ensemble – Set of repeated sampling procedures where statistics are applied across trial versus time or position domain
B. Types of Probability
- Rudimentary version: % of occurrences of particular events relative to total possible events
- Drawing 7 of spades
- Drawing 7 of ANY suit
- Roll of single die
- Doubles of pair of dice
- 3 consecutive doubles (jail in Monopoly)
- Combination of sets of outcomes
- Union and intersection – adding of new events that overlap, removing repeat counts
- Conditional probability – likelihood given prior event occurrence
- Permutation:
\[ P^n_m = \frac{n!}{(n - m)!} \]
Order matters - Combination:
\[ C^n_m = \frac{n!}{m!(n - m)!} \]
Order independent
C. Representing Statistical Information
- Sample vs. Population
True Value Sample Statistic \( x' \) \( \bar{x} \) \( \sigma^2 \) \( S^2_x \) - Histogram Representation: binning data and plotting frequency
- Frequency Distribution: normalized histogram converging to PDF as \( N \to \infty \)
D. Probability Density Function
- Must integrate to 1: \[ \int_{-\infty}^{+\infty} p(x) dx = 1 \]
- Expected value: \[ E[f(x)] = \int_{-\infty}^{+\infty} f(x)p(x)dx \]
E. Central Moments
- Zeroth: \( \mu_0 = 1 \)
- First: \( \mu_1 = 0 \)
- Second: \( \mu_2 = \sigma^2 \)
- Third: \( \mu_3 = \int (x - x')^3 p(x)dx \) — skewness
- Fourth: \( \mu_4 = \int (x - x')^4 p(x)dx \) — kurtosis
F. Standard PDFs
- Bernoulli, Binomial, Poisson, Weibull, Normal (Gaussian), Student t
- Gaussian PDF:
\[ p(x) = \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{(x - x')^2}{2\sigma^2}} \]
- Normalized variable: \( \beta = \frac{x - x'}{\sigma} \)
- Probability between values:
\[ P(x' - \delta x \leq x \leq x' + \delta x) = \int_{x' - \delta x}^{x' + \delta x} p(x) dx \]
G. Student t Distribution
- Sample mean: \[ \bar{x} = \frac{1}{N} \sum_{i=1}^N x_i \]
- Sample variance: \[ S_x^2 = \frac{1}{N - 1} \sum_{i=1}^N (x_i - \bar{x})^2 \]
- Normalized t-statistic:
\[ t = \frac{x - \bar{x}}{S_x} \]
- Prediction:
\[ x_{N+1} = \bar{x} \pm t_{\nu,P} S_x \]
H. Factorial Design
- Systematic variation of input parameters
- Main Effect:
\[ ME_j = \frac{1}{2^{k}} \sum_{i=1}^{2^k} (\pm r_i) \]
- Sensitivity:
\[ \zeta_{B} = \frac{ME_B}{B^+ - B^-} \]