W8-slides2 - Lecture 23- Outline and Examples Sums of...

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Unformatted text preview: Lecture 23- Outline and Examples Sums of independent random variables (Ross § 6.3) Sums of independent random variables Sums of independent r.v.s Let X and Y be independent r.v.s. Then their joint pdf is f X , Y ( x , y ) = f X ( x ) f Y ( y ) in continuous case or p X , Y ( x , y ) = p X ( x ) p Y ( y ) in discrete. Our goal is to compute distribution of Z = X + Y . Sums of independent random variables Sums of independent r.v.s Let X and Y be independent r.v.s. Then their joint pdf is f X , Y ( x , y ) = f X ( x ) f Y ( y ) in continuous case or p X , Y ( x , y ) = p X ( x ) p Y ( y ) in discrete. Our goal is to compute distribution of Z = X + Y . Discrete case: • p Z ( z ) = X x p X ( x ) p Y ( z- x ) Continuous case: • f Z ( z ) = Z ∞-∞ f X ( x ) f Y ( z- x ) dx (convolution integral) Example. If X ∼ Poisson ( λ ) and Y ∼ Poisson ( μ ) are independent then show that Z = X + Y ∼ Poisson ( λ + μ ). Example. If X ∼ Poisson ( λ ) and Y ∼ Poisson ( μ ) are independent then show that Z = X + Y ∼ Poisson ( λ + μ )....
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This note was uploaded on 06/05/2010 for the course STAT PStat 120a taught by Professor Rohinikumar during the Spring '10 term at UCSB.

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W8-slides2 - Lecture 23- Outline and Examples Sums of...

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