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notes509fall11sec32 - STAT 509 Section 3.2: Discrete Random...

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STAT 509 – Section 3.2: Discrete Random Variables Random Variable : A function that assigns numerical values to all the outcomes in the sample space. Notation : Capital letters (like Y ) denote a random variable. Lowercase letters (like y ) denote possible values of the random variable. Discrete Random Variable : A numerical r.v. that takes on a countable number of values (there are gaps in the range of possible values). Examples: 1. Number of phone calls received in a day by a company 2. Number of heads in 5 tosses of a coin Continuous Random Variable : A numerical r.v. that takes on an uncountable number of values (possible values lie in an unbroken interval). Examples: 1. Length of nails produced at a factory 2. Time in 100-meter dash for runners Other examples?
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The probability distribution of a random variable is a graph, table, or formula which tells what values the r.v. can take and the probability that it takes each of those values. Example 1: A design firm submits bids for four projects. Let Y = number of successful bids. y 0 1 2 3 4 P( y ) 0.06 0.35 0.43 0.15 0.01 Example 2: Toss 2 coins. The r.v. Y = number of heads showing. y 0 1 2 P( y ) ¼ ½ ¼ Graph for Example 1: For any probability distribution: (1) P( y ) is between 0 and 1 for any value of y . (2) y y P ) ( = 1. That is, the sum of the probabilities for all possible y values is 1.
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Example 3: P( y ) = y / 10 for y = 1, 2, 3, 4. Valid Probability Distribution? Property 1? Property 2? Cumulative Distribution Function: If Y is a random variable, then the cumulative distribution function (cdf) is denoted by F( y ). F( y ) = P(Y < y ) cdf for r.v. in Example 1? Graph of cdf:
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Expected Value of a Discrete Random Variable The expected value of a r.v. is its mean (i.e., the mean of its probability distribution). For a discrete r.v. Y , the expected value of Y , denoted μ or E( Y ), is: μ = E( Y ) = y y yP ) ( where y represents a summation over all values of Y . Recall Example 3: μ = Here, the expected value of y is Recall Example 1: What is the expected number of successful design bids? E(
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notes509fall11sec32 - STAT 509 Section 3.2: Discrete Random...

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