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Various probability distributions including bernoulli, binomial, geometric, negative binomial, poisson, hypergeometric, uniform, exponential, and normal distributions. It provides formulas for calculating probabilities and properties of each distribution. Examples are also included for understanding the concepts.

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Pre 2010

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Download Probability Distributions: Discrete and Continuous and more Study notes Mathematical Statistics in PDF only on Docsity! STAT/MATH 309 DISCUSSION 4 TA: Jingjiang(Jack) Peng Office: 1275 MSC, 1300 Universtiy Avenue E-mail: [email protected] Phone: 262-1577 Office Hour: 11:30-1:30 p.m. Tuesday or by appoitment Websit: www.stat.wisc.edu/∼ peng 1 Discrete Distribution • Bernoulli: P(X=1)=θ, P(X=0)=1-θ • Binomial: flip n coins, each of which has probability θ of coming up heads, and the probability 1 − θ of coming up tails. Let X be the total number of heads. Then X is binomial(n,θ). P (X = k) = ( n k ) θk(1 − θ)n−k, k = 0, 1, 2, . . . , n Remark: If X1, X2, . . . , Xn are independent idential distribution (i.i.d) of Bernoulli(θ), Then Y = X1 + X2 + · · · + Xn is Binomial(n,θ) • Geometric: X be the number of tails that appears before the first head. P (X = k) = θ(1 − θ)k , k=0,1,. . . • Negative binomial: Y be the number of tials that appear before the rth head. P (Y = k) = ( r + k − 1 r − 1 ) θr(1 − θ)k, k = 0, 1, 2, . . . • Possion(λ):P (Y = k) = λ ke−λ k! , k = 0, 1, 2, . . . • Hypergeometric (without replacement) P (X = k) = (Mk )( N−M n−k ) (Nn) 2 Continous Distribution • Uniform[L,R]: density function f(x) = 1 R−L , L ≤ x ≤ R. • Exp(λ): f(x) = λe−λx, x ≥ 0.P (X ≥ x) = e−λx One important property of expontial distribution is ”memoryless”: P (Y ≥ y + h|Y ≥ h) = P (Y ≥ y) See Problem 2.4.6 • Standard normal, N(0,1): f(x) = 1√ 2π e−x 2/2, x ∈ R • Normal N(µ, σ2): f(x) = 1√ 2π e− (x−µ)2 2σ2 , x ∈ R. 1