Chapter3_print - STAT511 Summer 2009 Lecture Notes 1...

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STAT511 — Summer 2009 Lecture Notes 1 Chapter 3: Discrete Random Variable and Probability Distribution June 21, 2009 3 Discrete Random Variables Chapter Overview Random Variable (r.v.) Definition Discrete and continuous r.v. Probability distribution for discrete r.v. Mass function Cumulative distribution function (CDF) Some discrete probability distribution Binomial Geometric and Hypergeometric Poisson Uniform Expectation and variance Expectation Variance Poisson Process 3.1 Random Variables Random Variable Definition 1. A random variable (r.v.) is a real valued function of the sample space S . It is any rule that associates a number with each outcome in S . i.e., a r.v. is some number associated with a random experiment. Notation: we use upper case letters X , Y , Z , · · · , to denote random variables. Lower case x will be used to denote different values of a random variable X . A random variable is a real valued function, so the expression: X ( s ) = x means that x is the value associated with the outcome s (in a specific sample space S ) by the r.v. X . Example 2 . Toss a coin, the sample space is S = { H, T } . Let X be the r.v. associated with this random experiment, and let X ( H ) = 1, X ( T ) = 0. Or we may simply write X = x, x = 0 , 1. Purdue University Chapter3˙print.tex; Last Modified: June 21, 2009 (W. Sharabati)
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STAT511 — Summer 2009 Lecture Notes 2 Random Variable Examples: Bernoulli rv Example 3.1.1 Check if a manufactured computer component is defect. If it is defect X = 1, if not X = 0. Example 3.1.2 Let X = 1 if the life of a light bulb is over 1000 hours, X = 0 if not. Definition 3. A Bernoulli rv has two possible values 0 and 1. A Bernoulli rv is like an “indicator” variable I : I ( A ) = 1 , If event A occurs; 0 , If event A doesn’t occur. Example 3.1.3 Toss a coin 3 times. Let I 1 be the Bernoulli variable for the first toss, I 2 be the Bernoulli variable for the second toss, I 3 be the Bernoulli variable for the third toss. I i = 1, if head, I i = 0 if tail; for i = 1 , 2 , 3. Let X be the totally number of heads tossed, we have: X = I 1 + I 2 + I 3 Types of Random Variables: Discrete & Continuous A discrete rv is an rv whose possible values constitute a finite set or a countably infinite set. A continuous rv is an rv whose possible values consists of an entire interval on the real line. Example 3.1.4 X = number of tosses needed before getting a head. Example 3.1.5 X = number of calls a receptionist gets in an hour. Example 3.1.6 X = life span of a light bulb. Example 3.1.7 X = weight of a Purdue female student. 3.2 Probability Distributions Probability Distribution for Discrete RVs Distribution of an rv, vaguely speaking, is how an rv distributes its probabilities on real numbers. For a discrete rv, we may list the values and the probability for each value of the rv, this gives the probability distribution of the discrete rv. Such rv is said to be an rv with discrete distribution .
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