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# lecture3 - ISYE2028 A and B Probability Lecture 3 Dr Kobi...

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ISYE2028 A and B: Probability Lecture 3 Dr. Kobi Abayomi January 15, 2009 1 Introduction: We work with Random Variables Remember that we use probability to describe outcomes of experiments 1 which we call events 2 If probability is the frame of statistics then Random Variables are the engine. We use random variables to link probability to data. Random variables are the numbers associated with events. 2 Definition of a Random Variable A random variable associates a numerical value with each outcome of an experiment. A random variable is a function from the sample space to real numbers. In notation: X : Ω R (1) Remember that Ω is the sample space of an experiment. Remember that a function, say f : A → B associates an element of A with an element of B . Lets look at a simple example. 1 flipping a coin, rolling a die, measuring a drug concentration in a rat... 2 getting a heads, seeing a 4, finding .01 percent of mercury in a rats body... 1

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2.1 Ex: Flipping a Die I flip a coin once, record the face. I flip a coin again, record the face. The sample space for the general experiment is the collection of all possible outcomes. Ω = { HH, TT, HT, TH } = { E 1 , E 2 , E 3 , E 4 } A probability distribution for an experiment is the enumeration of all possible outcomes with their associated probabilities. If I enumerate all outcomes separately (i.e. completely disjoint)... Event P { Event } E 1 1 4 E 2 1 4 E 3 1 4 E 4 1 4 Lets say that I am only interested in the number of heads in this experiment. Same experi- ment - but I view it differently: Event P { Event } E 1 1 4 E 2 1 4 E 3 or E 4 1 4 + 1 4 = 1 2 Often we do not recast the general experiment. Rather, we define the random variable appropriately: Let X the number of heads in two flips of a coin. Then, for this experiment, the probability distribution for the random variable is X P ( X ) 0 1 4 1 1 2 2 1 4 Thats the general way we do it: Regard the sample space completely and let the random variable be assigned to events appropriately. It is worth restating: The above table is the probability distribution for the random variable X . Notice that X is capitalized; if I write x I mean an observed value of the experiment, X means a possible value. So X refers to the population or world of probability and x refers to the sample or world of observed data. 2
2.2 Discrete Random variable We call a random variable discrete if it has a countable number of values. The above example is obviously discrete . A discrete random variable arises from an experiment which involves counting. Examples of discrete random variables are: number of heads in n coin flips; number of people who get flu; number of people who get a flu shot before one gets a reaction. Etc. Etc. 2.3 Probability Distribution of a Random Variable A list, table, or functional specification of the numerical values of X , a random variable, and the probabilities associated with it is the probability distribution of a random variable .

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lecture3 - ISYE2028 A and B Probability Lecture 3 Dr Kobi...

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