This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: APPENDIX B Fundamentals of probability 1. RANDOM VARIABLES An experiment is any procedure that can, at least in theory, be infinitely repeated and has a welldefined set of outcomes . E.g. flip a coin thrice and record the number of times the coin turns up heads. A random variable is a variable that takes on numerical values and the values that a random variable can take are determined by an experiment. E.g. in the coin flipping experiment, the number of heads that appear in 3 flips of a coin is an example of a random variable. Why is it called a random variable? Because before the experiment is conducted, the value of the random variable is unknown. It is only after conducting the experiment that we obtain the value/outcome of the random variable for that particular trial of the experiment. Another trial can produce a different outcome. Random variables are denoted by uppercase letters, usually W, X, Y and Z . Particular outcomes of random variables are denoted by the corresponding lowercase letters, w, x, y and z . 1. RANDOM VARIABLES The properties of a random variable can be described by its sample space and its probability distribution over the sample space. • The set of all possible outcomes/values for the random variable is called the sample space . The outcomes can be discrete or continuous. • The likelihood of a RV taking any given value in the sample space is governed by the probability distribution of the RV. The probability that a RV takes a particular value from the sample space has the following properties: • The probability that a RV takes any single value in the sample space has to be between 0 and 1. • All the probabilities must sum to 1. A random variable can be fully characterized by its probability density function (pdf). The pdf of a random variable summarizes the information concerning the possible outcomes of the RV and the corresponding probabilities. Its denoted by: ? ? = ? = (? = ? ) 1a. DISCRETE RANDOM VARIABLES A discrete random variable is one that takes on only a finite number of values. Simple example: Experiment: “Tossing a single coin” There are only 2 (i.e. discrete) possible outcomes: a head or a tail. • Assumption: it is a “fair” coin i.e. the probability that a head occurs is equal to the probability that a tail occurs and equal to 0.5. We define a random variable X : X=1 if the coin turns up heads, and X=0 if the coin turns up tails. Sample space : {0,1} Any particular outcome can be denoted by: x i : i = 1, 2 (since RV has two possible values/outcomes) 1a. DISCRETE RANDOM VARIABLES In this case, the probability density function (pdf) looks like: Random variable X= x i : Probability ( ) =p i = P(X= x i :) (pdf) x 1 = 0 (Tails) 0.5 x 2 = 1 (Heads) 0.5 ?...
View
Full
Document
This note was uploaded on 09/25/2011 for the course ECON 322 taught by Professor Francisco during the Spring '07 term at Rutgers.
 Spring '07
 Francisco

Click to edit the document details