What you'll learn
Unit 4 covers probability, random variables, and key distributions — the foundation for inference.
Probability rules
- 0 ≤ P(A) ≤ 1; complement: P(not A) = 1 − P(A).
- Addition: P(A or B) = P(A) + P(B) − P(A and B).
- Multiplication: P(A and B) = P(A)·P(B | A).
- Mutually exclusive events can't co-occur (P(A and B) = 0).
Conditional probability & independence
- P(A | B) = P(A and B) / P(B).
- Independent if P(A | B) = P(A), equivalently P(A and B) = P(A)·P(B).
Random variables
- Discrete vs continuous.
- Expected value E(X) = Σ x·P(x) (the long-run mean).
- Variance/SD measure spread.
- Linear transformations: adding a constant shifts the mean (not the SD); multiplying scales both. For sums of independent variables, means add and variances add (SDs do not).
Binomial distribution
Fixed number of trials n, two outcomes, constant p, independent trials. Mean = np; SD = √(np(1−p)). Use for 'exactly/at least k successes.'
Geometric distribution
Number of trials until the first success. Mean = 1/p.
Exam tips
- State independence/conditions before using a rule or model.
- For binomial, identify n, p, and what 'success' means.
- Remember variances (not SDs) add for independent variables.
Common mistakes
- Adding standard deviations instead of variances.
- Assuming events are independent without justification.
- Confusing mutually exclusive with independent (they're different).