Probability Distributions and Sampling (Basics)

1) Probability: core ideas

Probability measures the likelihood of an event.

If all outcomes are equally likely:

Properties:

• $0 \le P(E) \le 1$
• $P(S)=1$ (S = sample space)
• $P(E^c) = 1 - P(E)$

2) Random variable

A random variable assigns numerical values to outcomes of a random experiment.

Types:

• Discrete: takes countable values (0,1,2,...)
• Continuous: takes values over an interval (e.g., height, income in theory)

3) Probability distribution

A probability distribution describes probabilities of values of a random variable.

3.1 Discrete distribution

Given $X$ takes values $x_i$ with probability $p_i$:

Expected value (mean):

Variance:

4) Common distributions (basic recognition)

4.1 Binomial distribution (discrete)

Used when:

• fixed number of trials $n$,
• each trial has success probability $p$,
• trials independent,
• two outcomes: success/failure.