handout_week4_a - [ X + c ] = VAR [ X ] VAR [ cX ] = c 2...

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Stochastic Signals and Systems Random Variables Virginia Tech Fall 2008 Variance Mean alone will not be able to truly represent the pdf of any random variable. We need at least an additional parameter to measure the rv’s spread around the mean For a rv X with mean m , X - m represents the deviation of the rv from its mean. Since this deviation can be either positive or negative, consider the quantity ( X - m ) 2 , and its average value E [( X - m ) 2 ] represents the average square deviation of X around its mean. Define VAR [ X ] = E [( X - m ) 2 ]
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Variance By taking the square root of the variance we obtain a quantity with the same unit as X . The standard deviation of the rv X is defined by STD [ X ] = VAR [ X ] 1 / 2 The variance can be rewritten as VAR [ X ] = E [( X - m ) 2 ] = E [ X 2 - 2 E [ X ] X + E [ X ] 2 ] = E [ X 2 ] - E [ X ] 2 If X is a discrete rv, we find for its variance VAR [ X ] = X k ( x k - E [ X ]) p X ( x k ) Example Let the probability density function of a rv X be f X ( x ) = ± 1 4 + 1 2 δ ( x - 1 4 ) - 1 < x < 1 0 | x | > 1 Find the mean value and variance of X .
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Variance The following properties are easy to show VAR [ c ] = 0 VAR
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Unformatted text preview: [ X + c ] = VAR [ X ] VAR [ cX ] = c 2 VAR [ X ] The n th moment of the rv X is dened by E [ X n ] = Z - x n f X ( x ) dx The mean and variance can be seen to be dened in terms of the rst two moments, E [ X ] and E [ X 2 ] . Of greater signicance are the central moments E [( X-E [ X ]) n ] = Z - ( x-E [ X ]) n f X ( x ) dx Example Find the variance of a Gaussian random variable. Conditional Expected Values Expected values with respect to conditional probability density functions are called conditional expected values . If M is a conditioning event, E [ g ( X ) | A ] = Z - g ( x ) f X ( x | A ) dx is the conditional expected value of g ( X ) , given A . The conditional variance of a random variable X with respect to a conditioning event A is VAR [ X | A ] = E [ X 2 | A ]-E [ X | A ] 2...
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handout_week4_a - [ X + c ] = VAR [ X ] VAR [ cX ] = c 2...

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