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Chapter 4.1-4.3

# Chapter 4.1-4.3 - Statistics 511 Statistical Methods Dr...

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Statistics 511: Statistical Methods Dr. Levine Purdue University Spring 2011 Lecture 8: Continuous Random Variables: an Introduction Devore: Section 4.1-4.3 Feb, 2011 Page 1

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Statistics 511: Statistical Methods Dr. Levine Purdue University Spring 2011 Continuous Random Variables: a motivating example What probability distribution formalizes the notion of ”equally likely” outcomes in the unit interval [0 , 1] ? If we assign P ( X = 0 . 5) = ε for any real ε > 0 , we have a serious problem. Consider the event E = 1 2 , 1 3 , 1 4 , . . . , Then, P ( E ) = P j =2 1 j = X j =2 ε = We must assign a probability of zero to every outcome x in [0 , 1] Feb, 2011 Page 2
Statistics 511: Statistical Methods Dr. Levine Purdue University Spring 2011 Interpretation There is nothing shocking about it : an empty set (an impossible event) must have probability zero but nobody ever said that an event that has probability zero is always impossible... We also conclude that any countable event also has probability zero as well Moreover, if we think of ”equally likely” outcomes as meaning that an outcome is equally likely to be in two subintervals of equal length, we have 1 = P ( X [0 , 1]) = P ( X [0 , 0 . 5])+ P ( X [0 . 5 , 1]) - P ( X = 0 . 5) = P ( X [0 , 0 . 5])+ P ( X [0 . 5 , 1]) and, therefore P ( X [0 , 0 . 5]) = P ( X [0 . 5 , 1]) = 1 2 Feb, 2011 Page 3

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Statistics 511: Statistical Methods Dr. Levine Purdue University Spring 2011 Continuous uniform distribution Let S be the sample space, X ( S ) = [0 , 1] and each x [0 , 1] is equally likely. Then, for any 0 a b 1 P ( X [ a, b ]) = b - a This is called continuous uniform distribution . Its cdf is easy to compute: 1. If y < 0 , F ( y ) = P ( X y ) = 0 2. If y [0 , 1] , F ( y ) = P ( X y ) = P ( X [0 , y ]) = y 3. If y > 1 , F ( y ) = P ( X y ) = P ( X [0 , 1]) = 1 Feb, 2011 Page 4
Statistics 511: Statistical Methods Dr. Levine Purdue University Spring 2011 Continuous random variable: definition A random variable X is continuous if its set of possible values is an entire interval of numbers The function f is called a probability density function (pdf; compare to pmf) if f ( x ) 0 for any x R and R -∞ f ( x ) dx = 1 . A random variable is continuous if there exists a pdf f such that for any two numbers a and b, P ( a X b ) = Z b a f ( x ) dx For any two numbers a and b with a < b P ( a X b ) = P ( a < X < b ) = P ( a X < b ) = P ( a < X b ) .

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Chapter 4.1-4.3 - Statistics 511 Statistical Methods Dr...

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