PPT for Chapter 2 - Probability and Distribution

Slide 2: Chapter Overview

Title: Overview of Probability and Distribution Content:

  • Covers sample space, events, probability rules, conditional probability.
  • Includes sensitivity/specificity, discrete distributions, binomial, uniform, and normal distributions.
  • Explores mean, variance, and practical applications.

Slide 3: Sample Space (2.1)

Title: What is Sample Space? Content:

  • Definition: Set of all possible outcomes of a random experiment (denoted as S).
  • Examples:
    • Rolling a die: S = {1, 2, 3, 4, 5, 6}
    • Waiting time for a message: S = [0, ∞]
  • Types:
    • Finite (limited outcomes)
    • Infinite (countable or uncountable)

Slide 4: Events (2.2)

Title: Understanding Events Content:

  • Definition: Subset of sample space; specific outcome or set of outcomes.
  • Example: Die roll event “even number” = {2, 4, 6}
  • Types:
    • Simple event: Single outcome (e.g., rolling a 3)
    • Compound event: Multiple outcomes (e.g., even numbers)
  • Operations: Union (\(A\cup B\)), Intersection (\(A\cap B\)), Complement (\(A^c\) or \(\bar{A}\))

Slide 5: Probability Basics (2.3)

Title: Probability: Measuring Likelihood Content:

  • Definition: Likelihood of an event (0 = impossible, 1 = certain).
  • Formula: For equally likely outcomes, P(A) = (Outcomes in A) / (Total outcomes in S)
  • Example: Die roll, event “even number”: P(A) = 3/6 = 0.5

Slide 6: Basic Probability Rules (2.4)

Title: Key Probability Rules Content:

  • Non-negativity: \(P(A) ≥ 0\)
  • Normalization: \(P(S) = 1\)
  • Addition Rule (Mutually Exclusive): \(P(A∪B) = P(A) + P(B)\)
  • Complement Rule: \(P(A^c) = 1 - P(A)\)

Slide 7: Conditional Probability (2.5)

Title: Conditional Probability Content:

  • Definition: Probability of event A given event B has occurred, P(A|B).
  • Formula: \(P(A|B) = P(A∩B) / P(B)\)
  • Example: Deck of 52 cards, \(P(Heart|Red) = (13/52)/(26/52) = 1/2\)
  • Independence: \(P(A∩B) = P(A)·P(B)\) or \(P(A|B) = P(A)\)

Slide 8: Sensitivity and Specificity (2.6)

Title: Application: Sensitivity and Specificity Content:

  • Context: Medical test accuracy.
  • Sensitivity: P(Positive|Disease)
  • Specificity: P(Negative|No Disease)
  • Example: Test with sensitivity 0.95, specificity 0.90.
    • 95% diseased test positive.
    • 90% healthy test negative.

Slide 9: Discrete Distribution (2.7)

Title: Discrete Probability Distribution Content:

  • Definition: Probability for discrete random variables (countable values).
  • Probability Mass Function (PMF): P(X = x) for each value x.
  • Property: Sum of probabilities = 1.
  • Example: Daily complaints in a coffee shop (X: 0-3, Probabilities: 0.5, 0.3, 0.13, 0.07).

Slide 10: Mean and Variance of Random Variables (2.8)

Title: Mean and Variance Content:

  • Mean (Expected Value): \(E(X) = \Sigma [x · P(X = x)]\), central tendency.
  • Variance: \(Var(X) = \Sigma [(x - \mu)² · P(X = x)]\), measures spread.
  • Standard Deviation: \(\sigma = \sqrt{Var(X)}\)
  • Example (Die Roll): E(X) = 3.5, Var(X) ≈ 2.92, σ ≈ 1.71

Slide 11: Binomial Distribution (2.9)

Title: Binomial Distribution Content:

  • Definition: Number of successes in n independent trials, success probability \(p\).
  • Formula: \(P(X = k) = \frac{n!}{k!\cdot (n-k)!} · p^k · (1-p)^{n-k}\)
  • Mean and Variance: \(E(X) = n·p, ~Var(X) = n·p·(1-p)\)
  • Example: 10 coin flips (\(p=0.5\)), \(P(3~ \text{heads}) \approx 0.117\), \(P(6~ \text{heads}) \approx 0.205\)

Slide 12: Uniform Distribution (2.10)

Title: Uniform Distribution (Finite Interval) Content:

  • Definition: Continuous variable equally likely between \(a\) and \(b\).
  • Mean: \((a + b)/2\)
  • Standard Deviation: \((b - a)/\sqrt{12}\)
  • Example: Delivery time 20-40 mins, mean = 30 mins, P(25-30 mins) = 25%

Slide 13: Normal Distribution (2.11)

Title: Normal Distribution Content:

  • Definition: Bell-shaped, continuous, defined by mean (\(\mu\)) and standard deviation (SD: \(\sigma\)).

  • Empirical Rule:

    • 68% within 1 SD
    • 95% within 2 SDs
    • 99.7% within 3 SDs

  • Example: IQ scores (\(\mu=100\), \(\sigma=15\)), \(P(85-115) \approx 68\%\)

Slide 14: Conclusion (2.12)

Title: Chapter Summary Content:

  • Covered sample space, events, probability rules, conditional probability.
  • Explored sensitivity/specificity in testing.
  • Introduced distributions: discrete, binomial, uniform, normal.