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lecture%202_complete - APPENDIX B Fundamentals of...

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APPENDIX B Fundamentals of probability
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1. RANDOM VARIABLES An experiment is any procedure that can, at least in theory, be infinitely repeated and has a well-defined 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 .
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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: ? ? 𝑖 = ? 𝑖 = 𝑃(? = ? 𝑖 )
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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)
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1a. DISCRETE RANDOM VARIABLES In this case, the probability density function (pdf) looks like: Random variable X= x i : Probability 𝒇(𝒙 𝒊 ) =p i = P(X= x i :) (pdf) x 1 = 0 (Tails) 0.5 x 2 = 1 (Heads) 0.5 ?
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