Statistics and Probability: Lecture 1
Basic Concepts of Probability
Statistics is the collection and analysis of data. Data is a collection of outcomes from a series of random experiments. A random experiment is one with uncertain outcomes. Sample space is the set of all possible outcomes of a random experiment. A random variable is an unknown that can be equal to any value in the sample space. The distribution of a random variable describes the frequencies of that variable. The distribution of a random variable can be displayed graphically via a histogram. The mode is the value of a random variable that has the largest probability. Probability is the relative frequency (f/n) of an outcome when a random experiment is repeated a relatively large number of times. Probability models are used to describe how a random variable is distributed. An event is any subset of a sample space, and is a specific set of outcomes.
Properties of Probability
The probability of any event E must be on the interval [0, 1], so that 0 ≤ P(E) ≤ 1. If S is the sample space, then P(S) = 1. The probability of the union of mutually exclusive events equals the sum of the probability of each event, so that P(A ∪ B ∪ …) = P(A) + P(B) + …
The complementary rule says that P(A`) = 1 - P(A).
The probability of the null event is equal to 0, or P(∅) = 0.
If A ⊂ B, then P(A) ≤ P(B).
The addition rule says that P(A ∪ B) = P(A) + P(B) - P(A ∩ B).
Events are disjoint if they have no common outcomes.
P(A ∪ B ∪ C) = P(A) + P(B) + P(C) - P(A ∩ B) - P(A ∩ C) - P(B ∩ C) + P(A ∩ B ∩ C)