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Sum of Normally Distributed Random Variables

# Sum of Normally Distributed Random Variables - Sum of...

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Sum of Normally Distributed Random Variables This appendix provides a series of derivations for the density function of the sum of independent random variables. We begin with the simplest case, namely, the sum of two independent standard normal random variables 1 Z and 2 Z . The probability density functions for these random variables are defined as follows. () () 12 2 1 exp 22 ZZ z fz fz π  ==  For a normally distributed random variable X with mean X η and variance 2 X σ , the probability density function is given by () 2 2 2 1 exp 2 2 X X X X x fx x η σ πσ   = −− <<  In Chapter 3, it was shown that the density function for the sum of two independent random variables, WUV =+ , can be obtained from the convolution integral. () ( ) () WU V fw fwx d x −∞ = For the standard normal density function we obtain the following result. 2 2 2 2 2 11 exp exp 2 2 1 exp 2 2 2 exp exp 2 1 exp W wx x d x w x d x w xx w d x w ππ π −∞ −∞ −∞   =   = + = = 2 2 2 1 exp 24 4 exp exp 2 2 ww w d x xd x π −∞ −∞ +  = To complete the evaluation of the integral, we make the change of variable 2 u x = .

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() 2 2 2 2 2 11 1 exp exp 22 2 2 2 2 2 2 exp exp 2 2 1 exp W wu w fw d u uw w du w ππ π π π −∞ −∞   = −−    = = The integral in the next to the last line is value of the normal distribution function U F + where the associated random variable has mean 2 w and variance 1. The remaining function is the probability density function for a normally distributed random variable with mean 0 and variance 2. This consistent with the result obtained from the calculation of the expected value and variance for the sum of two standard normal random variables. Given that {} 1 1 0 Z EZ η == , 2 2 0 Z η , 2 2 1 1 ZZ ση = = , 2 2 2 1 = = , and ( ) ( ) 12 1 2 1 2 , Z Z fz z f z f z = , the expected value of the sum WZZ =+ and its variance can be derived as follows. ( ) () () () () 1 2 1 2 1 2 2 1 2 21 2 1 2 2 1 2 2 1 2 1 2 2 1 1 2 , ,, W Z Z EW EZ Z zz f z z d z d z z f z z dz dz z f z z dz dz z f z f z dz dz z f z f z dz dz zf z dz f z dz η ∞∞ −∞ −∞ −∞ −∞ −∞ −∞ −∞ −∞ −∞ −∞ −∞ −∞ −∞ = ∫∫ () () 0 EZ F F ηη −∞ = + = The variance is formally defined as follows
() {} 12 11 2 2 2 2 2 2 2 2 22 2 2 2 2 WW ZZ Z Z Z Z EW EZ Z EZ Z Z Z Z ση ηη η η σσσ =  =+ +   = + = + −− + + In Chapter 3, it was demonstrated that two independent random variables are uncorrelated and hence have a zero covariance. It follows that the variance for the sum

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Sum of Normally Distributed Random Variables - Sum of...

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