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ch7 - Chapter 7 Sums of Random Variables and Long-Term...

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Unformatted text preview: Chapter 7 Sums of Random Variables and Long-Term Averages ENCS6161 - Probability and Stochastic Processes Concordia University Sums of Random Variables Let X 1 , ··· ,X n be r.v.s and S n = X 1 + ··· + X n , then E [ S n ] = E [ X 1 ] + ··· + E [ X n ] V ar [ S n ] = V ar [ X 1 + ··· + X n ] = E n summationdisplay i =1 ( X i- μ X i ) n summationdisplay j =1 ( X j- μ X j ) = n summationdisplay i =1 V ar [ X i ] + n summationdisplay i =1 n summationdisplay j =1 i negationslash = j Cov ( X i ,X j ) If Z = X + Y ( n = 2) , V ar [ Z ] = V ar [ X ] + V ar [ Y ] + 2 Cov ( X,Y ) ENCS6161 – p.1/14 Sums of Random Variables Example : Sum of n i.i.d r.v.s with mean μ and variance σ 2 . E [ S n ] = E [ X 1 ] + ··· + E [ X n ] = nμ V ar [ S n ] = nV ar [ X i ] = nσ 2 pdf of sums of independent random variables X 1 , ··· ,X n indep r.v.s and S n = X 1 + ··· + X n , then Φ S n ( w ) = E [ e jwS n ] = E [ e jw ( X 1 + ··· + X n ) ] = Φ X 1 ( w ) ··· Φ X n ( w ) and f S n ( s ) = F- 1 { Φ X 1 ( w ) ··· Φ X n ( w ) } ENCS6161 – p.2/14 Sums of Random Variables Example : X 1 ··· X n indep and X i ∼ N ( m i ,σ 2 i ) . What is the pdf of S n = X 1 + ··· + X n ? For a Guassian r.v. X ∼ N ( μ,σ 2 ) ⇒ Φ X ( w ) = e jwμ- w 2 σ 2 2 (prove it by yourself) So Φ S n ( w ) = n productdisplay i =1 e jwm i- w 2 σ 2 i 2 = e jw ( m 1 + ··· + m n )- w 2 ( σ 2 1 + ··· + σ 2 n ) / 2 ∴ S n ∼...
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ch7 - Chapter 7 Sums of Random Variables and Long-Term...

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