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# LectureNotes4 - Discrete Random Variables and Their...

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Discrete Random Variables and Their Probability Distribution: Part I Cyr Emile M’LAN, Ph.D. [email protected] Discrete Random Variable: Part I – p. 1/40

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Introduction Text Reference : Introduction to Probability and Its Application, Chapter 4. Reading Assignment : Sections 4.1-4.2, February 16-February 18 In this chapter we extend the concepts and techniques of probability introduced in Chapter 2. We’ll also study how probability provides the basis for making statistical inferences. Suppose that one flips a coin 100 times and counts the number of heads. The objective is to infer from the count that the coin is not balanced. Discrete Random Variable: Part I – p. 2/40
Introduction It is reasonable to believe that observing a large number of heads (say, 90) or a small number of tails (say, 15) would be an indication of an unbalanced coin. However, where do one draw the line? At 85 or 75 or 65 or 55? Without knowing the probability of the frequency of the number of heads from a balanced coin, one cannot draw such a line. Hence, we would not be able to draw any conclusions from the sample of 100 coin flips. The concepts and techniques of probability developed in this chapter will allow us to calculate such thresholds we seek. Discrete Random Variable: Part I – p. 3/40

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Random Variables Sample space need not consist of numbers. But very often we are interested in numerical outcomes associated with the random phenomenon generating the sample space. Example 4.1 : Consider an experiment where we flip two balanced coins and observe the results. The sample space is S = { ( H, H ) , ( H, T ) , ( T, H ) , ( T, T ) } . However, one can list the events in a different way by counting the number of heads (or if we wish, the number of tails). Thus, the four simple events are replaced now by: 2 heads, 1 head, 1 head, 0 heads, resp. The number of heads, X , is called a random variable (r.v.). The induced sample space associated to the r.v. X is S star = { 0 , 1 , 2 } . Discrete Random Variable: Part I – p. 4/40
Random Variables Example 4.2 : In the game of craps played in casinos, the player tosses two dice. A natural way of listing the events is to describe the number on the first die and the number on the second die. Hence, the sample space is. S = { ( i, j ) , i, j = 1 , 2 , 3 , 4 , 5 , 6 } . However, in this game the player is primarily interested in the total sum of the two dice. If we define the random variable X to be the total of the two dice, X can take integer values between 2 and 12. Hence, the induced sample space for Y is S star = { 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 } . Discrete Random Variable: Part I – p. 5/40

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Random Variables Definition 4.1 : A random variable is a real-valued function or rule defined over a sample space that assigns a number to each outcome of a random experiment. Types of Random Variables There are two basic types of random variables: discrete and continuous .
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