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lecture4 - ISYE 2028 A and B Lecture 4 Dr Kobi Abayomi 1...

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ISYE 2028 A and B Lecture 4 Dr. Kobi Abayomi January 20, 2009 1 Introduction - Continuous Random Variables We call a random variable continuous if it has an uncountable number of values; if it can take all values in an interval of values. Examples of continuous random variables: Survival time of drinkers of Smoke-Cola r ; Time to recidivism for parolee of Savings and Loan Scandal; Amount of weight lost. Etc. Etc. That the definition of continuous closely matches the version we use in single variable calculus is natural and should make us feel good. We can extend what we’ve said already about discrete random variables, using ’s, to say analogous things about continuous random variables, using R ’s. Remember that the integral, R , is just the limit of , as the discrete index goes to be an infinitesimal. 1 . 2 Probability Distribution of a Random Variable Let’s extend the definition of the probability distribution to the continuous case by first restating that the distribution is the complete specification of values of the random variable with assigned probabilities . In the discrete case we could use this heuristic to write down a function or a table. In the continuous case, the distribution of the random variable is explicitly functional. 1 In Leibniz’s view of the calculus. Now would be a good time to break out your Calc I textbook, if you need to 1

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Here is an explicit definition of a continuous probability distribution or probability density function (pdf): For X a continuous random variable, the pdf of X is the function f ( X ) such that: P ( a X b ) = Z b a f ( x ) dx (1) we call, f ( x ), the density curve for X . We can also restate some of our probability rules using this new definition. For X a continuous random variable, on the real line, with density function f ( x ) R x -∞ f ( u ) du = F ( x ). F ( x ) is called the distribution function for X . F ( x ) = P ( X x ), or the probability that and random variable X is less than or equal to x . R + -∞ f ( x ) dx = F (+ ) - F ( -∞ ) = 1 - 0 = 1. Pay attention to nuance here: The distribution function of X is 1 at infinity. Every value of X is less than or equal to infinity. There is an analogous argument for F ( -∞ ) = 0. And I point out that the area under the density curve must equal 1. For all X ] - ∞ , + [, 0 f ( x ) 1 2.1 Example Say we have an interval A = { x : 0 x 2 } where we observe a real valued random variable, X . Say we believe the distribution function is of some form F ( x ) = cx 2 , with c a constant. We can immediately determine c : since F (2) = 1 = 4 c c = 1 / 4. As well, F ( x ) = x 2 4 = R 2 0 f ( u ) du f ( x ) = x 2 . Then the probabilities for any interval, for example P ( 1 4 X 1 2 ) = R 1 / 2 1 / 4 u 2 du = F (1 / 2) - F (1 / 4) = 3 / 64. 2.2 Features The property of the complement yields: F ( x ) P ( X > x ) = 1 - P ( X x ) = 1 - F ( x ) (2) 2
1 - F ( x ) is called the survival distribution for X .

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lecture4 - ISYE 2028 A and B Lecture 4 Dr Kobi Abayomi 1...

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