Normal Approximations to Binomial & Poisson Distributions

Normal Approximation: Big Picture

• Core idea: A continuous normal distribution can closely mimic discrete distributions (Binomial, Poisson) when certain conditions hold.
• Why useful?
  • Direct computation of many discrete probabilities is labor-intensive (hundreds or thousands of terms).
  • The normal CDF (and $Z$-tables/software) lets us evaluate areas quickly.
• Key strategy: Match the mean and standard deviation of the normal to those of the target discrete distribution.

Binomial → Normal Approximation

• Set-up
  • Binomial parameters: $n$ (number of trials), $p$ (success probability).
  • Corresponding normal: $\mu = np$, $\sigma = \sqrt{np(1-p)}$.
• Illustrative data-crunch problem
  • Survey of $n = 2000$ people, $p = 0.5$ believe the product is superior.
  • Desired probability: $P(X \le 600)$, where $X \sim \text{Bin}(2000, 0.5)$.
  • Direct method: must add 601 separate binomial terms $[P(X=0)+P(X=1)+\cdots+P(X=600)]$, each requiring $\displaystyle P(X=k)=\binom{2000}{k}0.5^{k}0.5^{2000-k}$ ▸ clearly impractical.

Concrete Example for Parameters

• If $n = 20$, $p = 0.5$:
  • $\mu = 20\times0.5 = 10$
  • $\sigma = \sqrt{20\times0.5\times0.5} \approx 2.2361$.
• A plotted histogram (bars) of the binomial probabilities aligns almost perfectly under the red bell curve of this normal distribution.

Rule-of-Thumb Conditions

• Reasonable approximation: $np \ge 5$ and $n(1-p) \ge 5$.
• Quite good approximation: $np \ge 10$ and $n(1-p) \ge 10$.
  ➔ Both the “successes” and “failures” should have expected counts of at least 5–10.

Continuity Correction (Essential Step)

Because the normal is continuous and the binomial is discrete:

• Adjust the cutoff by $\pm0.5$ to capture the full width of the bar representing each integer count.
• General recipe:
  • “$P(X=k)$” becomes “$P(k-0.5 \le Y \le k+0.5)$”.
  • “$P(X \le k)$” becomes “$P(Y \le k+0.5)$”.
  • “$P(X \ge k)$” becomes “$P(Y \ge k-0.5)$”.

Worked Mini-Example: $P(X = 5)$ with $n=20$, $p=0.5$

• Define $Y \sim N(\mu=10,\, \sigma\approx2.2361)$.
• Because $P(Y=5)=0$ for a continuous variable, use the band:
  $P(4.5 \le Y \le 5.5)$.
• Graphically: Shade the bell area between 4.5 and 5.5; this area approximates the height of the discrete bar at $X=5$.

Poisson → Normal Approximation

• Poisson parameters: Mean (and variance) $\lambda$.
• Matching normal: $\mu = \lambda$, $\sigma = \sqrt{\lambda}$.
• Accuracy notes:
  • Gets better as $\lambda$ increases.
  • Poor when \lambda < 5. (Below that, stick to exact Poisson or other approximations.)
• Continuity correction still required for the same reason as with the binomial.

Practical & Conceptual Takeaways

• Approximations convert tedious summations into one or two normal CDF look-ups.
• Ensuring the conditions ($np, n(1-p)$ or $\lambda$ large enough) and applying the continuity correction prevent systematic error.
• Insight: The Central Limit Theorem underlies why discrete sums approach a bell curve, justifying all such normal approximations.