Probability Distributions
A distribution describes how probabilities are distributed over the values of a random variable.
Discrete Distributions
For variables that can be counted (e.g., number of customers).
Binomial Distribution
Number of successes in $n$ trials.
$$ P(x) = \binom{n}{x} p^x (1-p)^{n-x} $$
Poisson Distribution
Number of events in a fixed interval (time/space).
$$ P(x) = \frac{e^{-\lambda} \lambda^x}{x!} $$
Continuous Distributions
For variables that can be measured (e.g., time, weight).
Probability is area under the curve.
The Normal Distribution (Bell Curve)
The most important distribution in statistics. Symmetric, bell-shaped.
- 68% of data within $\pm 1\sigma$
- 95% of data within $\pm 2\sigma$
- 99.7% of data within $\pm 3\sigma$
Standard Normal (Z)
Mean = 0, Std Dev = 1.
$$ Z = \frac{X - \mu}{\sigma} $$
Test Yourself
Q1: Which distribution models the number of arrivals at a bank per hour?