Statistics/Probability
Introduction to probability
Please note that this page is just a stub, more will be added later.
Why have probability in a statistics textbook?
Very little in mathematics is truly self contained. Many branches of mathematics touch and interact with one another, and the fields of probability and statistics are no different. A basic understanding of probability is vital in grasping basic statistics, and probability is largely abstract without statistics to determine the "real world" probabilities.
This section is not meant to give a comprehensive lecture in probability, but rather simply touch on the basics that are needed for this class, covering the basics of Bayesian Analysis for those students who are looking for something a little more interesting. This knowledge will be invaluable in attempting to understand the mathematics involved in various Distributions that come later.
Set notion
A set is a collection of objects. We usually use capital letters to denote sets, for e.g., A is the set of females in this room.
• The members of a set A are called the elements of A. For e.g., Patricia is an element of A (Patricia ∈ A) Patrick is not an element of A (Patrick ∉ A).
• The universal set, U, is the set of all objects under consideration. For e.g., U is the set of all people in this room.
• The null set or empty set, ∅, has no elements. For e.g., the set of males above 2.8m tall in this room is an empty set.
• The complement Ac of a set A is the set of elements in U outside A. I.e. x ∈ Ac iff x ∉ A.
• Let A and B be 2 sets. A is a subset of B if each element of A is also an element of B. Write A ⊂ B. For e.g., The set of females wearing metal frame glasses in this room ⊂ the set of females wearing glasses in this room ⊂ the set of females in this room.
• The intersection A ∩ B of two sets A and B is the set of the common elements. I.e. x ∈ A ∩ B iff x ∈ A and x ∈ B.
• The union A ∪ B of two sets A and B is the set of all elements from A or B. I.e. x ∈ A ∪ B iff x ∈ A or x ∈ B.
Venn diagrams and notation
Probability
Probability is connected with some unpredictability. We know what outcomes may occur, but not exactly which one. The set of possible outcomes plays a basic role. We call it the sample space and indicate it by S. Elements of S are called outcomes. In rolling a dice the sample space is S = {1,2,3,4,5,6}. Not only do we speak of the outcomes, but also about events, sets of outcomes. E.g. in rolling a dice we can ask whether the outcome was an even number, which means asking after the event "even" = E = {2,4,6}. In simple situations with a finite number of outcomes, we assign to each outcome s (∈ S) its probability (of occurrence) p(s) (written with a small p), a number between 0 and 1. It is a quite simple function, called the probability function, with the only further property that the total of all the probabilities sum up to 1. Also for events A do we speak of their probability P(A) (written with a capital P), which is simply the total of the probabilities of the outcomes in A. For a fair dice p(s) = 1/6 for each outcome s and P("even") = P(E) = 1/6+1/6+1/6 = 1/2.
The general concept of probability for non-finite sample spaces is a little more complex, although it rests on the same ideas.
Negation
Negation is a way of saying "not A", hence saying that the complement of A has occurred. For example: "What is the probability that a six-sided die will not land on a one?" (five out of six, or p = 0.833)
Or, more colloquially, "the probability of 'not X' together with the probability of 'X' equals one or 100%."