notes515fall10chap4(1)

notes515fall10chap4(1) - STAT 515 - Chapter 4: Discrete...

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STAT 515 -- Chapter 4: Discrete Random Variables Random Variable : A variable whose value is the numerical outcome of an experiment or random phenomenon. 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? 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.
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Example 1: Roll 1 die. The r.v. X = number of dots showing. x 1 2 3 4 5 6 P( x ) 1/6 1/6 1/6 1/6 1/6 1/6 Example 2: Toss 2 coins. The r.v. X = number of heads showing. x 0 1 2 P( x ) ¼ ½ ¼ Graph for Example 2: For any probability distribution: (1) P( x ) is between 0 and 1 for any value of x . (2) x x P ) ( = 1. That is, the sum of the probabilities for all possible x values is 1. Example 3: P( x ) = x / 10 for x = 1, 2, 3, 4. Valid Probability Distribution? Property 1? Property 2?
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The expected value of a r.v. is its mean (i.e., the mean of its probability distribution). For a discrete r.v. X , the expected value of X , denoted or E( X ), is: = E( X ) = x P( x ) where represents a summation over all values of x . Recall Example 3: = Here, the expected value of X is Example 4: Suppose a raffle ticket costs $1. Two tickets will win prizes: First prize = $500 and second prize = $300. Suppose 1500 tickets are sold. What is
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notes515fall10chap4(1) - STAT 515 - Chapter 4: Discrete...

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