AP Precalculus: Binomial & Normal Distributions Formulas
1. Binomial Distribution
- Discrete probability for \( n \) trials, probability of \( k \) successes:
-
\[
P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}
\]
- \( n \): number of trials
- \( k \): number of successes
- \( p \): probability of success
2. Mean, Variance, SD (Binomial)
- Mean: \( \mu = np \)
- Variance: \( \sigma^2 = np(1-p) \)
- Standard deviation: \( \sigma = \sqrt{np(1-p)} \)
3. Normal Distribution
- Continuous, bell-shaped probability curve
- Standard normal: mean \( 0 \), SD \( 1 \)
- Density function: \[ f(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{1}{2}\left( \frac{x - \mu}{\sigma} \right)^2} \]
4. Z-Scores & Using Normal Tables
- Z-score: \( z = \frac{x - \mu}{\sigma} \)
- Given \( z \), find probability from normal table (P chart or calculator)
- Given area/probability, find \( x \):
- \( x = \mu + z\sigma \)
5. Distributions of Sample Means
- Mean of sampling distribution: \( \mu_{\bar{x}} = \mu \)
- Standard deviation: \( \sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}} \), where \( n \) is sample size
6. Central Limit Theorem (CLT)
- For large \( n \), the distribution of sample means is approximately normal, regardless of original distribution
- Mean \( \mu \), standard deviation \( \frac{\sigma}{\sqrt{n}} \)
7. Normal Approximation to Binomial
- If \( np \geq 10 \) and \( n(1-p) \geq 10 \): binomial can be approximated by normal
- Use:
- Mean \( \mu = np \)
- Standard deviation \( \sigma = \sqrt{np(1-p)} \)
- Apply continuity correction (+/- 0.5 to bounds)