Approximations: Binomial, Poisson, and Normal

In Further Statistics, you must know when and how to approximate one distribution using another. These approximations simplify calculations and are essential when exact computation is impractical. This lesson covers the three key approximations: Poisson approximation to the binomial, normal approximation to the binomial, and normal approximation to the Poisson.

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Poisson Approximation to the Binomial

If $X \sim B(n, p)$X∼B(n,p) with:

• $n$n large (typically $n \geq 50$n≥50)
• $p$p small (typically $p \leq 0.1$p≤0.1)
• $np$np moderate (typically $np < 5$np<5)

then $X$X can be approximated by $\text{Po}(\lambda)$Po(λ) where $\lambda = np$λ=np.

Binomial	Poisson approximation
$B(n, p)$B(n,p)	$\text{Po}(np)$Po(np)
$E(X) = np$E(X)=np	$E(X) = \lambda = np$E(X)=λ=np
$\text{Var}(X) = np(1-p)$Var(X)=np(1−p)	$\text{Var}(X) = \lambda = np$Var(X)=λ=np

The approximation improves as $n \to \infty$n→∞ and $p \to 0$p→0 with $np$np held constant.

Why the variance approximation works: When $p$p is small, $1 - p \approx 1$1−p≈1, so $np(1-p) \approx np = \lambda$np(1−p)≈np=λ.

Worked Example

A manufacturer produces components with a defect rate of 2%. In a batch of 200, find the probability that exactly 3 are defective.