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Unformatted text preview: Chapter 3 General Random Variables 1 Continuous Random Variables and PDFs A random variable X is continuous if there is a nonnegative f X , called the probability density function of X (PDF), such that P [ a X b ] = Z b a f X ( x ) dx If X is a continuous RV with PDF f X , then For any x , P [ X = x ] = 0, and for any a and b , P [ a X b ] = P [ a < X b ] = P [ a X < b ] = P [ a < X < b ]. Z  f X ( x ) dx = 1. If is very small P [ x < X < x + ] f X ( x ) . Consider a continuous random variable whose PDF is given by f X ( x ) = cx 2 x 1 otherwise * Find c . * Compute P [ X < 1 / 2], P [ X 2] and more generally P [ X x ]. Let X be a continuous RV with PDF f X , then 1 2 MTH2222 Mathematics of Uncertainty E [ X ] = Z  xf X ( x ) dx * What is E [ X ] if f X ( x ) = 3 x 2 , 0 x 1. E [ g ( X )] = Z  g ( x ) f X ( x ) dx var( X ) = E [ X 2 ] E [ X ] 2 = Z  x 2 f X ( x ) dx Z  xf X ( x ) dx 2 * What is var( X ) if f X ( x ) = 3 x 2 , 0 x 1. var( X ) = E [( X E [ X ]) 2 ] 0, E [ aX + b ] = a E [ X ] + b and var( aX ) = a 2 var( X ). The Continuous Uniform Random Variable The PDF is f X ( x ) = 1 b a , a x b . The mean is E [ X ] = a + b 2 . The variance is var( X ) = ( b a ) 2 12 . The Exponential Random Variable The PDF is f X ( x ) = e x , x 0. The mean is E [ X ] = 1 . The variance is var( X ) = 1 2 . The Cauchy Random Variable Chapter 3 3 The PDF is f X ( x ) = ( x 2 + 2 ) . The mean and variance do not exist. 2 Cumulative Distribution Functions The cumulative distribution function (CDF) of a random variable X is F X ( x ) = P [ X x ] Draw the CDF of a binomial random variable with parameters n = 3 and p = . 5. Draw the CDF of an arbitrary discrete random variable (taking an arbitrary but finite number of of values). Draw the CDF of a uniform random variable over the interval [0 , 1]. Draw the CDF of an arbitrary continuous random variable. 4 MTH2222 Mathematics of Uncertainty The Geometric Random Variable The CDF is F X ( x ) = 1 (1 p ) b x c , x 0, where b x c is the integer part of x ....
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This note was uploaded on 10/09/2010 for the course MTH 2222 taught by Professor Kaizhamza during the Two '10 term at Monash.
 Two '10
 KaizHamza
 Probability

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