Chapter 2. measure c X is called the counting measure on X: Given a2 X;the probability measure a de–ned by the equation a(A) = ˜ A (a);if A2 P(X);is called the Dirac measure at the point a:Sometimes we write a = X;a to emphasize the set X: If and are positive measures de–ned on the same ˙-algebra M, the Analogously, a set in a measure space is said to have a σ-finite measure if it is a countable union of sets with finite measure.. For example, the real numbers with the standard Lebesgue measure are σ-finite but not finite. Outcomes, Counting, Measures and Probability 1 Outline Experiments, Outcomes, Events Counting – Finite Outcomes » Number of distinct strings, sequences, subsets and multisets – Infinite but Countable Outcomes – Uncountable Outcomes Probability Prof. Metin Çakanyıldırımused various resources to prepare this document for teaching/training. 1.1 Total variation distance Let Bdenote the class of Borel sets. An introductory example 31 2.

2.The probability that a selection of 6 numbers wins the National Lottery Lotto jackpot is 1 in 49 6 =13,983,816, or 7:15112 10 8.

A measure μ is called σ-finite if X can be decomposed into a countable union of measurable sets of finite measure. Discrete Probability Measures 33 3. equi-probability 38 Chapter 3.

1 Distances between probability measures Stein’s method often gives bounds on how close distributions are to each other. 1.The probability that a fair coin will land heads is 1=2.

Random Variables 55 1. Bayes Formula 44 3.

In probability theory and statistics, a probability mass function (pmf) is a function that gives the probability that a discrete random variable is exactly equal to some value. A typical distance between probability measures is of the type d( ; ) = sup ˆZ fd Z fd : f2D ˙; where Dis some class of functions. We analyse the complexity of sampling, of computing the normalizing factor, and of computing the size support of these distributions. Introduction to Probability Measures 31 1. In this paper we present a family of probability distributions over the set of solutions of a problem in TotP, and show how they relate to counting; counting is equivalent to computing their normalizing factors.

The probability mass function is often the primary means of defining a discrete probability distribution, and such functions exist for either scalar or multivariate random variables whose domain is discrete.

The notion of conditional probability 41 2. 3.The probability that a drawing pin will land ‘point up’ is 0:62.

Conditional Probability and Independence 41 1. Independence 50 Chapter 4.

4.The probability that a … Introduction 55 2. Introduction to Counting & Probability By David Patrick Learn the basics of counting and probability from former USA Mathematical Olympiad winner David Patrick.