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Lecture Notes 7

# Lecture Notes 7 - Chapter 5.Random Variables A random...

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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: Experiment Sample space Event Random Variable 1) Coin toss S={ H,T} A= “H” let X -# of heads , X(A) = 1 , possible values for X : x=0,1 2) Two-dice toss , S= {(1,1)(1,2)……. .} Let X- be the two dice total X((1,1))=1+1 =2 , x= 2,3,……12 3) Testing 8 elderly adults for the allergic reaction (yes or no) S- we have 256 possible outcomes ,for instance the event A=(yes,no,yes,no,no,no,yes,no) . Let X- # of allergic reactions among the set of eight adults . x=0,1,2,3,4,5,6,7,8 X(A)=3 Random variables can be discrete (if it has either a finite number of values or infinitely many values that can be arranged in a sequence) , or continuous(measurements on a continuous scale). The probability distribution for a discrete random variable 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 a) What is the probability that at least one is a subscriber? b) What is the probability that at most 1 is a subscriber? We can use the cumulative distribution function : a function that specifies, for each value x, the probability that X x. 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 replacement.Let X =the sum of the numbers on the selected marbles.Find the probability distribution of X .Find the cumulative distribution of X. Expectation and Variance
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Lecture Notes 7 - Chapter 5.Random Variables A random...

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