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chap3 - Chapter 3 General Random Variables 1 Continuous...

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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 δ is very small P [ x < X < x + δ ] f X ( x ) δ . Consider a continuous random variable whose PDF is given by f X ( x ) = cx 2 0 x 1 0 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
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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 ) = λe - λx , x 0. The mean is E [ X ] = 1 λ . The variance is var( X ) = 1 λ 2 . The Cauchy Random Variable
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Chapter 3 3 The PDF is f X ( x ) = α π ( x 2 + α 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.
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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 . – The Continuous Uniform Random Variable The CDF is F X ( x ) = x - a b - a , a x b . – The Exponential Random Variable The CDF is F X ( x ) = 1 - e - λx , x 0.
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