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2.1.1 Introduction to Random Distribution

1. Definition and Types of Random Variables

Discrete Random Variables

  • Example: Tossing a coin
  • Diagram: Probability tree showing outcomes (Heads/Tails)
  • Table: Probabilities of outcomes ( p and 1-p )
  • Example: Counting heads in 10 coin tosses
  • Table: Possible outcomes (0-10 heads) with corresponding probabilities

Continuous Random Variables

  • Example: Time until first head appears in coin tosses
  • Diagram: Plot showing probability distribution over time

2. Distribution Functions

  • Diagram: Graphs of typical discrete and continuous distribution functions

Important Definitions

  1. Probability Mass Function (PMF): For discrete variables
  2. Probability Density Function (PDF): For continuous variables

3. Expected Value, Variance, and Standard Deviation

Expected Value

  • Formula: E[X] = Sum(x * P(x))
  • Example Calculation: Using coin toss data

Variance and Standard Deviation

  • Formulas: Var(X), SD(X)
  • Example Calculation: Using coin toss data
  • Diagram: Visual explanation of variance in distribution

4. Linear Combinations of Random Variables

  • Explanation: What it means to have linear combinations
  • Formula: General form and specific example
  • Example: Combining two dice rolls
  • Table: Possible outcomes and their probabilities

5. Practical Applications of Random Variables

  • Real-world relevance of random variables
  • Examples: Business contexts (not provided in transcript, suggest potential applications)
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