Module 2-1_Random Variables-Expectation-Dispersion-Skewness 2025

Module Overview

  • Course Title: ENCI 604 Uncertainty, Risk and Reliability

  • Module Title: Module 2 Random Variables and Probability Distributions

  • Key Topics:

    • Introduction to Random Variables

    • Expectation, Dispersion, Skewness

    • Discrete Probability Distribution

    • Continuous Probability Distribution

    • Poisson Process

Introduction to Random Variables

  • Key Concepts:

    • Random Variables (r.v.) map outcomes from a sample space into the real number line.

    • Notation: Random variable denoted as X; the value taken by X denoted as x.

  • Types of R.V.: Several random variables can exist on the same sample space.

Probability Distributions

Discrete vs Continuous Probability Distributions

Discrete Probability Distribution

  • Probability Mass Function (PMF):

    • Describes the probability of discrete random variables.

    • Notation: 𝑃𝑋(π‘₯) = 𝑃(𝑋 = π‘₯)

    • Properties:

      • 𝑃𝑋(π‘₯) β‰₯ 0

      • Sum of all probabilities equals 1: Οƒπ‘₯ 𝑃𝑋(π‘₯) = 1

Continuous Probability Distribution

  • Probability Density Function (PDF):

    • Describes the probability for continuous random variables.

    • Notation: Average values derived from integrals.

    • Properties:

      • Area under the curve = 1

      • Probability between two values: 𝑃(π‘Ž < π‘₯ < 𝑏) = βˆ«α΅‡β‚π‘“π‘‹(π‘₯)𝑑π‘₯

Comparison of PMF and PDF

  • PMF deals with discrete data while PDF deals with continuous data.

  • Cumulative Distribution Function (CDF) also exists for both types of distributions to summarize the total probability.

Descriptive Statistics

Central Tendency and Dispersion

  • Measures of Central Tendency:

    • Mean, Median, Mode

  • Variability Measures: Variance and Standard Deviation to assess data spread.

    • Notation for Variance: Var(X) = E(X - ΞΌ)Β²

Applications of Descriptive Statistics

  • Use Cases: Provide basic understanding of large data sets using software like Excel, R, SPSS.

Sampling Concepts

Sample vs Population

  • Population: All units of interest (finite/infinite).

  • Sample: A subset drawn from the population for analysis.

Statistical Calculations

Sample Mean, Median, Mode

  • Calculating:

    • Mean: Average of all observations.

    • Median: The middle value in the ordered set.

    • Mode: The most frequently occurring value.

Variance and Standard Deviation

  • Variance assesses variability within a sample.

  • Standard deviation is the square root of variance, representing average deviation from the mean.

Detailed Example of Descriptive Statistics

  • Worked example illustrating the calculation of mean, median, mode, variance, and standard deviation based on measured product weights.

Visualization Tools

Importance of Data Visualization

  • Tools include histograms, box plots, scatter diagrams, and pie charts for effective presentation and analysis of data.

Histograms

  • Used to approximate the probability distribution by displaying frequency of observations in defined intervals (bins).

  • Cumulative histograms aggregate frequencies across bins.

Box Plots

  • Excellent for comparing distributions across variables, indicating median, quartiles, and potential outliers.

Summary

  • Module emphasized the understanding and application of random variables, probability distributions, and descriptive statistics within data analysis.