Lecture Notes on Binomial Distribution

Introduction to Binomial Distribution

• Overview of binomial formula
  • Formula: P(X=k) = C(n,k) * p^k * (1-p)^(n-k)
  • Variables:
    • k: Number of successes
    • n: Number of trials
    • p: Probability of success

Example: Coin Flips

• Scenario: Flipping a coin 2 times
• Values:
  • k = 0, 1, 2 (number of successes)
  • n = 2 (number of trials)
  • p = 0.5 (probability of heads)
• Calculated Probabilities:
  • P(X=0) = 0.25
  • P(X=1) = 0.50
  • P(X=2) = 0.25

Visualization

• Bar Chart:
  • X-axis: Number of successes (0, 1, 2)
  • Y-axis: Probability
  • Observations:
    • P(0 successes) = 0.25
    • P(1 success) = 0.5
    • P(2 successes) = 0.25

Increasing Trials (n=10)

• Shape Change: Distribution resembles a normal distribution as n increases
• Mean of distribution: Centered around 5

Binomial Distribution Parameters

• Mean (μ): μ = n * p
• Variance (σ²): Variance = n * p * (1 - p)
• Standard Deviation (σ): σ = √(Variance)

Effect of Changing p

• p = 0.5 (n=10): Distribution is symmetrical
• Decreasing p (p=0.1):
  • Skewed left; less success expected
• Increasing p (p=0.8):
  • Skewed right; more success expected

Summary of Distribution Shapes

• p < 0.5: Skewed right (less success)
• p > 0.5: Skewed left (more success)
• p = 0.5: Symmetrical distribution

Normal Approximation of Binomial Distribution

• Conditions for approximation:
  1. n * p >= 10
  2. n * (1 - p) >= 10
• Rough guideline: Some use n * p >= 5 instead of 10

Recap of Key Points

• p controls the distribution shape
• Symmetrical when p = 0.5
• Skewed when p deviates from 0.5
• As n increases, binomial approaches normal distribution
• Mean (μ), Variance, and Standard Deviation formulas

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