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7: Probability and Statistics

  • Page ID
    121980
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    To further improve communication of experimental results, the combination of probability (expected outcomes) with statistics (methodology to reduce data) are covered.

    Learning Objectives
    • 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
    1. Probability – likelihood of particular event occurrence
    2. Statistics – post processing analysis tool for interpretation of probability
    3. Samples – discrete data points of collection in statistics that indicate probability
    4. Ensemble – Set of repeated sampling procedures where statistics are applied across trial versus time or position domain
    B. Types of Probability
    1. 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)
    2. 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
    1. Sample vs. Population
      True Value Sample Statistic
      \( x' \) \( \bar{x} \)
      \( \sigma^2 \) \( S^2_x \)
    2. Histogram Representation: binning data and plotting frequency
    3. 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:

      \[ \zeta_{B} = \frac{ME_B}{B^+ - B^-} \]


    7: Probability and Statistics is shared under a CC BY-NC 4.0 license and was authored, remixed, and/or curated by LibreTexts.

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