handout 7 - Chapter 5.Random Variables. A random variable X...

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1 Chapter 5.Random Variables. • A random variable X associates a numerical value with each outcome of an experiment. (It is frequently the case when an experiment is performed that we are mainly interested in some function of the outcome as opposed to the actual outcome itself). Examples: Examples: Experiment Sample space Event Random Experiment Sample space Event Random Variable Variable 1) 1) Coin toss S={ H,T} A= “H” let X Coin toss S={ H,T} A= “H” let X -# of heads , # of heads , H(A) = 1 , possible values for X : x=0,1 H(A) = 1 , for X : 2) 2) Two Two -dice toss ,S= {(1,1)(1,2)……. .} Let X dice toss ,S= {(1,1)(1,2)……. .} Let X - the two dice total the two dice total X((1,1))=1+1 =2 x= 2,3,……12 x= 2,3,……12 3) Testing 8 elderly adult for the allergic reaction (yes or n 3) Testing 8 elderly adult for the allergic reaction (yes or n o) o) S- we have 256 possible outcomes ,for instance : we have 256 possible outcomes ,for instance : A=(yes,no,yes,no,no,no,yes,no) . A=(yes,no,yes,no,no,no,yes,no) . Let X Let X - # of allergic reactions among set of eight adults . # of allergic reactions among set of eight adults . x=0,1,2,3,4,5,6,7,8 X(A)=3 X(A)=3 Random variables can be discrete (if it has either a finite number of values or infinitely many values finite number of values or infinitely many values that can be arranged in a sequence) , that can be arranged in a sequence) , or continuous(measurements on a continuous scale). continuous(measurements on a continuous scale). The probability distribution for a discrete random probability distribution for a discrete random variable variable X X is a graph, table or formula that gives the possible values of x and the probability f(x)=P(X=x) associated with each value. Properties of a probability distribution: 1) 0 f(x i ) 1; 2) 2) f(x i ) =1. Example 1 • Toss a fair coin three times and define X = number of heads. 1/8 1/8 1/8 1/8 1/8 1/8 1/8 1/8 P(X = 0) = 1/8 P( X = 1) = 3/8 P( X = 2) = 3/8 P( X = 3) = 1/8 HHH HHT HTH THH HTT THT TTH TTT X 3 2 2 2 1 1 1 0 1/8 3 3/8 2 3/8 1 1/8 0 f(x) x Probability Histogram for X Cumulative Distribution Function • What is the probability that at least one is a subscriber? • Can use the cumulative distribution function : a function that specify, for each value x, the probability that X x. • What is the probability that at most 1 is a subscriber? F(x)=P(X x) = p(X=x 1 )+p(X=x 2 )+….p(X=x k ) with x k x < x k+1 X f(x) 0 .49 1 .42 2 .09 Exercise 1. Let X is the number of subscribers to a magazine in a sample of 2 X f(x) F(x) 0 .49 .49 1 .42 .91 2 .09 1.00
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2 Exercise 2 From the six marbles numbered : 1,1,1,1,2,2 Two marbles will be drawn at random without
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This note was uploaded on 04/17/2008 for the course HONORS 191 taught by Professor Palmero during the Spring '08 term at UMass (Amherst).

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handout 7 - Chapter 5.Random Variables. A random variable X...

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