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Geometric Random Variables

Geometric Random Variables. N ~ Geometric(p) # Bernoulli trials until the first success pmf: f(k) = (1-p) k-1 p memoryless: P(N=n+k | N>n) = P(N=k) probability that we must wait k more coin flips for the first success is independent of n, the number of trials that have occurred so far.

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Geometric Random Variables

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  1. Geometric Random Variables N ~ Geometric(p) • # Bernoulli trials until the first success • pmf: f(k) = (1-p)k-1p • memoryless: P(N=n+k | N>n) = P(N=k) • probability that we must wait k more coin flips for the first success is independent of n,the number of trials that have occurred so far

  2. Previously… • Conditional Probability • Independence • Probability Trees • Discrete Random Variables • Bernoulli • Binomial • Geometric

  3. Agenda • Poisson • Continuous random variables: • Uniform, Exponential • E, Var • Central Limit Theorem, Normal

  4. Poisson N ~ Poisson() • N = # events in a certain time period • average rate is  • Ex. cars arrivals at a stop sign • average rate is 20/hr • Poisson(5) = #arrivals in a 15 min period

  5. Poisson • pmf: P(N=k) = e- k/k! • Excel: POISSON(k,,TRUE/FALSE) =3 =12.5

  6. Poisson N1~Poisson(1), N2~Poisson(2) • N1+N2 ~ Poisson(1+ 2) • Splitting: • Poisson() people arrive at L-stop • probability p person is south bound • Poisson(p) people arrive at L-stop south bound

  7. other slides… from Prof. Daskin’s slides

  8. Xrandom variable E[g(X)]=∑k g(k) P(X=k) E[a X+b] = aE[X] +b Var[a X + b] = a2 Var[X] always X1,…,Xn random variables E[X1+…+ Xn] = E[X1]+…+E[Xn] always Var[X1+…+ Xn] = Var[X1]+…+Var[Xn] when independent E[X1·X2·…· Xn] = E[X1]·E[X2] ·…·E[Xn] when independent E and Var

  9. X~Bernoulli(p) E[X]=p, Var[X]=p(1-p) X~Binomial(N,p) E[X]=Np, Var[X]=Np(1-p) N~Geometric(p) E[N]=1/p, Var[N]=(1-p)/p2 N~Poisson() E[N]= , Var[N]=  X~U[a,b] E[X]=(a+b)/2, Var[X]=(b-a)2/12 X~Exponential() E[X]=1/, Var[X]=1/2 E, Var

  10. Central Limit Theorem X1,…,Xn i.i.d, µ=E[X1], 2=Var[X1] • independent, identically distributed Sn = X1,…,Xn • E[Sn]=nµ, Var[Sn] = n2 • distribution approaches shape of Normal • Normal(nµ,n2)

  11. =1 =2 =4 Normal Distribution mean=0

  12. Normal Distribution X1 ~ N(µ1,12), X2 ~ N(µ2,22) • X1+X2 ~ N(µ1+µ2,12+22) • pdf, cdf NORMALDIST(x,µ,,TRUE/FALSE) • fractile / inverse cdf • p=P(X≤z) • NORMINV(p,µ,)

  13. Newsvendor Problem • must decide how many newspapers to buy before you know the day’s demand • q = #of newspapers to buy • b = contribution per newspaper sold • c = loss per unsold newspaper • random variable D demand

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