Slides19 - Lecture Stat 302 Introduction to Probability Slides 19 AD April 2010 AD April 2010 1 14 Sum of Independent Random Variables Consider Z =

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Lecture Stat 302 Introduction to Probability - Slides 19 AD April 2010 AD () April 2010 1 / 14
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Sum of Independent Random Variables Consider Z = X + Y where X and Y are disrete r.v. of respective p.m.f. p X ( x ) and p Y ( y ) then p Z ( z ) = y p X ( z y ) p Y ( y ) . Consider Z = X + Y where X and Y are continuous r.v. of respective p.d.f. f X ( x ) and f Y ( y ) then f Z ( z ) = Z f X ( z y ) f Y ( y ) dy . AD () April 2010 2 / 14
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Sum of Exponential Random Variables Consider two independent exponential r.v. X , Y of parameter λ (i.e. f X ( x ) = f Y ( x ) = λ e λ x 1 [ 0 , ) ( x ) ). The pdf of Z = X + Y is f Z ( z ) = 0 for z < 0 and for z > 0 f Z ( z ) = Z λ e λ ( z y ) 1 [ 0 , ) ( z y ) λ e λ y 1 [ 0 , ) ( y ) dy = λ 2 e λ z Z 0 1 [ 0 , ) ( z y ) dy = λ 2 e λ z Z z 0 1 [ 0 , ) ( z y ) dy = λ 2 ze λ z . AD () April 2010 3 / 14
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Sum of Gaussian Random Variables Consider two independent normal standard r.v. X , Y (i.e. f X ( x ) = f Y ( x ) = 1 p 2 π e x 2 / 2 ) then the pdf of Z = X + Y is f Z ( z ) = 1 2 π Z e ( z y ) 2 / 2 e y 2 / 2 dy where ( z y ) 2 + y 2 = z 2 + 2 y 2 2 yz = z 2 + 2 ( y z / 2 ) 2 z 2 / 2 = 2 ( y z / 2 ) 2 + z 2 / 2 So we have f Z ( z ) = e z 2 / 4 2 π p π Z 1 p π e ( y z / 2 ) 2 dy = e z 2 / 4 p 2 π p 2 Hence Z is a normal r.v. of mean 0 and variance 2.
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This note was uploaded on 10/21/2010 for the course STAT Stat302 taught by Professor 222 during the Spring '10 term at The University of British Columbia.

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Slides19 - Lecture Stat 302 Introduction to Probability Slides 19 AD April 2010 AD April 2010 1 14 Sum of Independent Random Variables Consider Z =

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