NotesOct6

# NotesOct6 - The Poisson random variable is the limiting...

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The Poisson random variable is the limiting case of the Binomial random variable as: n → ∞ ; p 0; np λ > 0. Notation : X Poiss( λ ). p X ( k ) = e - λ λ k k ! , k = 0 , 1 , 2 , . . . .

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Reproducing property of Poisson random variables Let X 1 , X 2 , . . . , X n be independent Poisson ran- dom variables with parameters λ 1 , λ 2 , . . . , λ n . Then the sum Y = X 1 + X 2 + . . . + X n has the Poisson distribution with parameter λ 1 + λ 2 + . . . + λ n .
Review of the important continuous random variables 1. Continuous models on a bounded interval The Uniform random variable takes values in a bounded interval ( a, b ), and has a constant density f X ( x ) = ( 1 b - a if a x b 0 otherwise. Notation : X U( a, b ).

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The Beta radom variable offers flexible shapes of a density of a bounded interval. A random variable X has the Beta distribution on [0 , 1] with parameters α > 0 and β > 0 if it has the density f X ( x ) = ( 1 B ( α,β ) x α - 1 (1 - x ) β - 1 if 0 < x < 1 0 otherwise. Notation : X Beta( α, β ).
B ( α, β ) is the Beta function : B ( α, β ) = Z 1 0 x α - 1 (1 - x ) β - 1 dx = Γ( α )Γ( β ) Γ( α + β ) , α, β > 0.

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• '10
• SAMORODNITSKY
• Probability theory, Poisson random variables, exponential random variable

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