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Lecture5standard - Statistics 511: Statistical Methods Dr....

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Unformatted text preview: Statistics 511: Statistical Methods Dr. Levine Purdue University Spring 2011 Lecture 5: Discrete Random Variables, Distributions and Moments Devore: Section 3.1-3.3 Feb, 2011 Page 1 Statistics 511: Statistical Methods Dr. Levine Purdue University Spring 2011 Random Variables For a given sample space S , a random variable (RV) is any mapping Y : S R . Essentially, it is a function whose domain is the sample space and whose range is R . It is also possible to consider complex-valued random variables. We will not do it in the current course, however. Feb, 2011 Page 2 Statistics 511: Statistical Methods Dr. Levine Purdue University Spring 2011 A set is denumerable if and only if its elements can be placed in one-to-one correspondence with natural numbers. A set is countable if and only if it is either finite or denumerable. Example of a denumerable set : a set of all even natural numbers...Why? 2 1 , 4 2 , 6 3 , 8 4 ,... Another example: a set of all integers...Indeed, ...,- 3 7 ,- 2 5 ,- 1 3 , 1 , 1 2 , 2 4 , 3 6 , 4 8 ,... Feb, 2011 Page 3 Statistics 511: Statistical Methods Dr. Levine Purdue University Spring 2011 A discrete random variable is a RV whose possible values make up a countable sequence. Ex.(Discrete) A person attempts to log on to a power-sharing system; the out- come is either a success, coded by 1 , or failure coded by zero. Thus, with S = { S,F } , we have X ( S ) = 1 , X ( F ) = 0 . Any RV that only takes values or 1 is called Bernoulli RV, in honor of Jacob Bernoulli (1654- 1705) . Feb, 2011 Page 4 Statistics 511: Statistical Methods Dr. Levine Purdue University Spring 2011 Example The quality control process: we sample batteries (or any other industrially man- ufactured product) as it comes off the conveyor line. Let F denote the faulty and S the good one. The sample space is S = { S,FS,FFS,... } . Let X be the number of batteries that is examined before the experiment stops. The, X ( S ) = 1 ,X ( FS ) = 2 ,... . Feb, 2011 Page 5 Statistics 511: Statistical Methods Dr. Levine Purdue University Spring 2011 Example of a non-discrete random variable Consider measuring the elevation above the sea level of a randomly chosen point within the continental US map (in feet). It will be- 282 y 14 , 494 where the left bound corresponds to the Death Valley and the right one to Mt. Whitney. This random variable is continuous. Feb, 2011 Page 6 Statistics 511: Statistical Methods Dr. Levine Purdue University Spring 2011 Probability Distributions The probability distribution or probability mass function (pmf) of a discrete RV is defined for every number x as p ( x ) = P ( X = x ) = P ( all s S :...
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Lecture5standard - Statistics 511: Statistical Methods Dr....

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