Probability, Random Variables, and Probability Distributions

Notes

Two categorical variables

A two-way table shows counts for two variables. Marginal relative frequencies use row or column totals; conditional relative frequencies look within one row or column. If conditional distributions differ, the variables are associated.

Probability rules

• $0 \le P(A) \le 1$, and $P(A^c) = 1 - P(A)$.
• Addition: $P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)$. Mutually exclusive events have $P(A \text{ and } B) = 0$.
• Conditional: $\displaystyle P(A \mid B) = \frac{P(A \text{ and } B)}{P(B)}$.
• Independent events: $P(A \mid B) = P(A)$, which means $P(A \text{ and } B) = P(A)P(B)$.
• Simulation: use random numbers to model a chance process and estimate probabilities.

Discrete random variables

• Mean (expected value): $\displaystyle \mu_X = \sum x_i p_i$. Standard deviation: $\displaystyle \sigma_X = \sqrt{\sum (x_i - \mu_X)^2 p_i}$.
• Binomial setting: fixed number of trials $n$, two outcomes, independent trials, same probability $p$. $P(X = k) = \binom{n}{k}p^k(1 - p)^{n-k}$, $\mu = np$, $\displaystyle \sigma = \sqrt{np(1 - p)}$.

Normal distributions

• The empirical rule: about 68%, 95% and 99.7% of values fall within 1, 2 and 3 standard deviations of the mean.
• Use $\displaystyle z = \frac{x - \mu}{\sigma}$ and the standard normal distribution for probabilities and percentiles.

Sampling distributions

A sampling distribution is the distribution of a statistic over all possible samples.

• $\hat p$: mean $p$, standard deviation $\displaystyle \sqrt{\frac{p(1 - p)}{n}}$, approximately normal when $np \ge 10$ and $n(1 - p) \ge 10$.
• $\bar x$: mean $\mu$, standard deviation $\displaystyle \frac{\sigma}{\sqrt n}$. The central limit theorem says $\bar x$ is approximately normal when $n$ is large (often $n \ge 30$), whatever the population shape.
• A statistic is unbiased if its sampling distribution is centered at the parameter. Bigger samples mean less variability.