05_01 - Continuous Random Variables p. 5-1 Recall: For...

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p. 5-1 Continuous Random Variables • Recall : For discrete random variables, only a finite or countably infinite number of possible values with positive probability. Often, there is interest in random variables that can take (at least theoretically) on an uncountable number of possible values, e.g., the weight of a randomly selected person in a population, the length of time that a randomly selected light bulb works, the error in experimentally measuring the speed of light. Example (Uniform Spinner, LNp.2-14): = (  , ] For ( a , b ] , P (( a , b ]) = b a /(2 ) Consider the random variables: X : R , and X ( ) = for  Y : R , and Y ( ) = tan ( ) for  Then, X and Y are random variables that takes on an uncountable number of possible values. p. 5-2 • Probability Density Function and Continuous Random Variable Definition. A function f : R R is called a probability density function (pdf) if 1. f ( x ) 0, for all x ( , ), and 2. −∞ f ( x ) dx =1 . Definition: A random variable X is called continuous if there exists a pdf f such that for any set B of real numbers P X ({ X B }) = B f ( x ) dx . For example, Notice that: P X ({ X = x })=0, for any x R , But, for  a < b , P X ({ X ( a , b ]})= P (( a , b ]) = b a /(2 ) > 0. Q : Can we still define a probability mass function for X ? If not, what can play a similar role like pmf for X ?
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This note was uploaded on 02/15/2012 for the course MATH 2810 taught by Professor Shao-weicheng during the Fall '11 term at National Tsing Hua University, China.

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05_01 - Continuous Random Variables p. 5-1 Recall: For...

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