http://www.csie.ntu.edu.tw/~hil/prob/11BernoulliBinomialPoisson2

2The Bernoulli Family

3

Bernoulli random variable4We say that X is a Bernoulli random variable with parameter p with 0 p 1 ifP(X = 1) = p;P(X = 0) = 1 p.

Q: Such an X is a discrete random variable, because _______________________.

Probability mass function56Expectation7Variance: 8Binomial random variable9We say that X is a binomial random variable with parameter (n, p) if X is the sum X1 + X2 + + Xn of n independent random variables X1, X2, , Xn , where each Xi is a Bernoulli random variable with parameter p.

Q: Such an X is a discrete random variable, because _______________________.

Q: X = n * Y, where Y is a Bernoulli random variable with parameter p ?

Probability mass function10 --- p=0.5x x1/2n11(i.e., ) = Expectation12Variance13 ?14

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Simon Denis Poisson1781 184017

Poisson random variable18: 19Q: ?

Poisson2021Variance22Poissonbinomial23PoissonbinomialnpPoisson with parameter np binomial with parameter (n,p)

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Poisson26nnXE[X] = Var(X) = 1

Xbinomial random variablePoisson random variable Y with parameter 1approximate: E[Y] = Var(Y) = 127

27(slide 05)

2829n = 20Poissonp(20,0) ~ 0.367879p(20,1) ~ 0.367879p(20,2) ~ 0.183940p(20,3) ~ 0.061313p(20,4) ~ 0.015328p(20,5) ~ 0.003066p(20,6) ~ 0.000511

0.3678794410.3678794410.1839397210.06131324020.015328310.003065662010.00051094366830birthday2(n)n2.nbirthday2(n)>0.5?31

3132Poisson33Poisson34birthday3(n)n3.nbirthday3(n)>0.5?35

35..36

Poisson37Poisson38

393940

40Poisson41

Poisson42

EiEj43

Poisson44

Poisson(n = 20, p = 0.5)45