Discrete_RVs

# M n m x nx px x n 0 n if x 0 1 2 3

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Unformatted text preview: n ≤ M ) Mass function: M N −M ( x )( n−x ) pX (x ) = (N ) 0 n if x = 0, 1, 2, 3, . . . , n otherwise Statistics: µ = E (X ) = n M N σ 2 = Var(X ) = n M 1 − N Math 30530 (Fall 2012) M N N −n n −1 Discrete Random Variables October 7, 2012 7 / 10 Hypergeometric(M , N , n) is like Binomial(n, M /N ) Viewing hypergeometric as “pick n, one after another”, both are X1 + X2 + . . . + Xn with each Xi ∼ Bernoulli(M /N ) Difference For Binomial(n, M /N ), Xi ’s are independent For Hypergeometric(M , N , n), they are not; Pr(Xn = 1) varies between M M − (n − 1) and N − (n − 1 ) N − (n − 1) depending on previous choices If n small compared to N , M , not much difference here Example: When polling 1000 people (without replacement) out of 100,000,000 to see who they will vote for, can model situation with Binomial (easy) rather than Hypergeometric (harder) Math 30530 (Fall 2012) Discrete Random Variables October 7, 2012 8 / 10 The Poisson process Events occur re...
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