lecture+9 - Discrete Distributions An Introduction Discrete...

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Discrete Distributions An Introduction
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Discrete vs. Continuous Distributions Discrete distributions – constructed from discrete (individually distinct) random variables. The domain of the discrete distribution is finite and countable. Note spelling: discrete not discreet Continuous distributions – based on continuous random variables. While a continuous distribution may have finite lower and upper bounds, an infinite set of values is possible (just add more significant digits!)
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Discrete vs. Continuous Distributions Random Variable a variable which contains the outcomes of a chance experiment; it is a mapping from experimental space to probability space. think of a random variable as a numeration of the outcomes (sample space) of an experiment We might have a random variable (call it X) that counts the number of heads obtained by flipping two fair coins simultaneously. Then values for the random variable X could be {0, 1, 2}. And we could map probabilities to X P(X=0) = .25, P(X=1) = .5
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Describing a Discrete Distribution A discrete distribution can be described by constructing a graph of the distribution. Measures of central tendency and variability can be applied to discrete distributions. 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 P(X) Probability
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Pick the Proper Discrete Distribution X P(X) -1 0 1 2 3 .1 .2 .4 .2 .1 1.0 Y P(Y) -1 0 1 2 3 -.1 .3 .4 .3 .1 1.0 Z P(Z) -1 0 1 2 3 .1 .3 .4 .3 .1 1.2 :
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Describing a Discrete Distribution Mean of discrete distribution – is the long run average . If the process is repeated long enough, the average of the outcomes will approach the long-run average (mean). Requires the process to eventually have a number which is the product of many processes (e.g., flipping two coins repeatedly). Mean of a discrete distribution In our example, μ = 0 *.25 + 1*0.5 + 2 *0.25 = 0 + 0.5 + 0.5 = 1.0. We would expect to have a long-run average of
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Describing a Discrete Distribution Variance of a discrete distribution are solved by using X in a manner similar to computing these values for grouped data. In our experiment… So, what is the standard deviation?
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The Binomial and Poisson To be or not to be…That is a binomial!
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Review: Probability Distribution Experiment: Toss a coin two times. Observe the number of heads. The possible results are: zero heads, one head, two heads. What is the probability distribution for the number of heads? X Counts the Sample Space First Second Number of heads 1 T T 0 2 T H 1 3 H T 1 4 H H 2 X P(X=x) 0 1/4 1 1/2
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10 Review: Random Variables Random variable Quantity resulting from an experiment that, by chance, can assume different values.
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lecture+9 - Discrete Distributions An Introduction Discrete...

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