Chapter5_DiscreteProbabilityDistributions

Chapter5_DiscreteProbabilityDistributions - Chapter 5...

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Chapter 5 Discrete Probability Distributions
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Discrete Random Variables Random variables A numerical value assigned to each outcome (or simple event) of an experiment, which is subject to change from trial-to-trial. Discrete Continuous Examples: No. of people flying on a given airline No. of ounces in a bottle of pop No. of defective parts in a machine Amount of time it takes to assemble a product
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Probability Distribution Discrete Random Variables Probability Distribution The prob. dist. of a random variable, X, is the distribution of probabilities associated with each value of X. Can be provided in the form of a table, graph, mathematical formula, etc. Example Toss a coin n times Let X = number of heads X is discrete with possible outcomes of 0, 1, 2, …, n Probability of the outcomes: P(X=0), P(X=1), …, P(X=n) The collection of probabilities represent the prob. dist. of X
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Discrete Random Variables Properties 1. The probability associated with every value of X lies between 0 and 1, inclusive 0 f(x i ) 1 1. The sum of the probabilities for all values of X is equal to 1. That is, f(x i )=1. 2. The probabilities are additive. That is, f(X=x i or x j ) = f(X=x i ) + f(X=x j )
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Discrete Random Variables Example Toss a coin 3 times Let X = number of heads (H) Recall the simple events: HHH, HHT, HTH, HTT, THT, TTH, THH, TTT Possible outcomes of X are 0, 1, 2, and 3 We can calculate probabilities for each outcome: f(X=0)=1/8, f(X=1)=3/8, f(X=2)=3/8, and f(X=3)=1/8
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Discrete Random Variables Example (cont) The events and probabilities can be displayed in a prob. dist. Table Let X = Number of heads
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Discrete Random Variables Expected Value Characteristics: Central tendency (location) and dispersion (variability) Mean If X is a discrete random variable, then the expectation (or Expected Value ) of X is given by μ = E(X) = Xf(X)
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Discrete Random Variables Expected Value Definition consistent with sample mean, which is x i /n Example: Suppose we have discrete r.v. X that can assume values of 0, 1, 2 with probability distribution f(X) = ¼, ½, ¼, respectively.
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Expected Value Relative Frequency Distribution 0 0.1 0.2 0.3 0.4 0.5 0.6 P(X) 0 1 2
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Expected Value (cont) Suppose we conducted the expt. 4 million times and observed X Find the mean of X… From the figure, we would expect approx. 1 million to be X=0, 2 million X=1, and 1 million X=2.
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Expected Value (cont) Averaging the 4 million measurements, we get = = + + + + μ 2 0 x i 1 ) x ( xP ) 4 / 1 )( 2 ( ) 2 / 1 )( 1 ( ) 4 / 1 )( 0 ( 000 , 000 , 4 ) 2 )( 000 , 000 , 1 ( ) 1 )( 000 , 000 , 2 ( ) 0 )( 000 , 000 , 1 ( n x
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Discrete Random Variables Variance
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Example Let the random variable X be the grade received in a course. The units of are points. X f(X)
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This note was uploaded on 04/02/2008 for the course BIT 2405 taught by Professor Plkitchin during the Spring '08 term at Virginia Tech.

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Chapter5_DiscreteProbabilityDistributions - Chapter 5...

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