Normal Approximation to Binomial

Probability · Axiom Academy

LESSON Normal Approximation to Binomial Using the Central Limit Theorem to approximate binomial probabilities 1. Why the Normal Approximation Works A binomial random variable X ~ B(n, p) can be written as a sum of independent Bernoulli trials: where each Xᵢ ~ Bernoulli(p). Since X is a sum of i.i.d. random variables, the CLT applies! For large n: 2. When Can We Use the Approximation? Rule of Thumb: The normal approximation is good when: This ensures the distribution is neither too skewed nor concentrated at the extremes. 3. How to Apply the Approximation Step 1: Check that np ≥ 5 and n(1-p) ≥ 5 Step 2: Compute μ = np and σ = √(np(1-p)) Step 3: Standardize and use the normal table: Example: Suppose X ~ B(100, 0.3). Find P(X ≤ 35). Check: np = 30 ≥ 5 ✓ and n(1-p) = 70 ≥ 5 ✓ Parameters: μ = 30, σ = √21 ≈ 4.58 Approximate: P(X ≤ 35) ≈ P(Z ≤ (35-30)/4.58) = P(Z ≤ 1.09) ≈ 0.862

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