MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN: 9781119256830
Author: Amos Gilat
Publisher: John Wiley & Sons Inc
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3. A volunteer is distributing flyers at a street corner for a comedy show. The number of people who accept a flyer as they pass by follows a Poisson distribution with the rate of µ = 4 people per hour. a) What is the probability that it takes more than 10 minutes for the volunteer to give out the first flyer? e b) What is the expected number of flyers the volunteer would distribute in 6 hours? c) Suppose the volunteer has 3 flyers left. What is the exptected time it would take to run out of flyers? d) What is the probability that it would take more than an hour to distribute the last 3 flyers?
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Transcribed Image Text:3. A volunteer is distributing flyers at a street corner for a comedy show. The number of people who accept a flyer as they pass by follows a Poisson distribution with the rate of µ = 4 people per hour. a) What is the probability that it takes more than 10 minutes for the volunteer to give out the first flyer? e b) What is the expected number of flyers the volunteer would distribute in 6 hours? c) Suppose the volunteer has 3 flyers left. What is the exptected time it would take to run out of flyers? d) What is the probability that it would take more than an hour to distribute the last 3 flyers?
Axloms of Probablity Also Note 1. P(8)-1 2. For any event E, 0S P(E)s1 For any two events A and B, P(A) - P(AN B) + P(ANB) 3. For any two mutually exclusive events, and P(EUF) - P(E) + P(F) P(AN B) - P(A|B)P(B). Addition Rule Events A and B are Independent if: P(EUF) = P(E) + P(F) - P(En F) P(A|B) = P(A) Conditional Probablity or P(B|A) - P(ANB) - P(A)P(B). PLAN) Bayes' Theorem: Total Probablity Rule P(A|B)P(B) P(B|A) = PLALBPB) + P(AB)P(B") P(A) - P(A|B)P(B) + P(A|B')P(B') Similarly, Similarly, P(A) -P(A|E,)P(E)) + P(A|E)P(E)+ ...+ P(A|E)P(E) P(B|E)P(E) P(E|B) - PIBIE PE) + P(BE PE)+...+ P(B\E)P(E.) Probability Mass and Density Functions If X is a discrete r.v: Cumulative Distribution Function • F(z) = P(X sz) P(X = 2) = f(z) • lim,- F() -0 Es(2) =1 (total probability) • lim,e F(z) = 1 If X is a continuous r.v.: P(X = z) = 0 • F(z) = " /(v)dy if X is a contimuous r.v. S(2)dz =1 (total probability) • F(z) = E,sz f(z) if X is a discrete r.v. • P(a < X Sb) - F(b) – F(a) Expected Value and Variance Expected Value of a Function of a RV • E[X) = E, z/(z) if X is a discrete r.v. • E[h(X)] =E. h(x)f(x) if X is a discrete r.v. • Eh(X)) = h(z)/(z)dz if X is a continu- • E[X] = z/(z)dr if X is a continuous r.v. ous r.v. • Var(X) = E[Xx] – E[X]? • E(aX + 6) = aEX] + 6 • Var(aX + b) = a?Var(X) %3D • Var(X) = E[(X - E[X])?] Derivatives and Integrals of Common Functions • = aea de • Sea" dz = • Sre*dr = e"I- fe*dz = ze" - e (using integration by parts) dinz • S !dz = In(z) Common Discrete Distributions • X - Bernoulli(p), if z = 1; f(z) = |1-p ifz 0' EX] = p, Var(X) = p(1 – p). • X- Geometric(p), f(2) = (1– p)--'p, z E {1,2,..}, E[X] = }, Var(X) = . Geometric Series: Eg = , for 0 < q < 1 • X - Binomial(n, p), f(z) = (E) (1– p)"-p*, I € {0, 1,.., n}, E[X] = np, Var(X) = mp(1 – p). %3D • X- Negative Binomial(r, p), f(z) = ()(1 – p)*-"p", E[X] = ;, 1 € {r,r+1,..}, Var(X) = p), %3D • X - Hypergeometric(n, M, N), f(z) = , E[X] = n, Var(X) = N=n(1-). %3D • X ~ Poisson(At), f(z) = A0", z e {0, 1, .}, E[X] = At, Var(X) = At. Common Continuous Distributions • X - Exponential(A), f(z) = de-A, z E [0, 00) E[X] = }, Var(X)= . • X- Erlang(r, A), f(z) = A' , zE (0, 00), E[X] = 5, Var(X) = . Suppose that Duke Energy mu
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Transcribed Image Text:Axloms of Probablity Also Note 1. P(8)-1 2. For any event E, 0S P(E)s1 For any two events A and B, P(A) - P(AN B) + P(ANB) 3. For any two mutually exclusive events, and P(EUF) - P(E) + P(F) P(AN B) - P(A|B)P(B). Addition Rule Events A and B are Independent if: P(EUF) = P(E) + P(F) - P(En F) P(A|B) = P(A) Conditional Probablity or P(B|A) - P(ANB) - P(A)P(B). PLAN) Bayes' Theorem: Total Probablity Rule P(A|B)P(B) P(B|A) = PLALBPB) + P(AB)P(B") P(A) - P(A|B)P(B) + P(A|B')P(B') Similarly, Similarly, P(A) -P(A|E,)P(E)) + P(A|E)P(E)+ ...+ P(A|E)P(E) P(B|E)P(E) P(E|B) - PIBIE PE) + P(BE PE)+...+ P(B\E)P(E.) Probability Mass and Density Functions If X is a discrete r.v: Cumulative Distribution Function • F(z) = P(X sz) P(X = 2) = f(z) • lim,- F() -0 Es(2) =1 (total probability) • lim,e F(z) = 1 If X is a continuous r.v.: P(X = z) = 0 • F(z) = " /(v)dy if X is a contimuous r.v. S(2)dz =1 (total probability) • F(z) = E,sz f(z) if X is a discrete r.v. • P(a < X Sb) - F(b) – F(a) Expected Value and Variance Expected Value of a Function of a RV • E[X) = E, z/(z) if X is a discrete r.v. • E[h(X)] =E. h(x)f(x) if X is a discrete r.v. • Eh(X)) = h(z)/(z)dz if X is a continu- • E[X] = z/(z)dr if X is a continuous r.v. ous r.v. • Var(X) = E[Xx] – E[X]? • E(aX + 6) = aEX] + 6 • Var(aX + b) = a?Var(X) %3D • Var(X) = E[(X - E[X])?] Derivatives and Integrals of Common Functions • = aea de • Sea" dz = • Sre*dr = e"I- fe*dz = ze" - e (using integration by parts) dinz • S !dz = In(z) Common Discrete Distributions • X - Bernoulli(p), if z = 1; f(z) = |1-p ifz 0' EX] = p, Var(X) = p(1 – p). • X- Geometric(p), f(2) = (1– p)--'p, z E {1,2,..}, E[X] = }, Var(X) = . Geometric Series: Eg = , for 0 < q < 1 • X - Binomial(n, p), f(z) = (E) (1– p)"-p*, I € {0, 1,.., n}, E[X] = np, Var(X) = mp(1 – p). %3D • X- Negative Binomial(r, p), f(z) = ()(1 – p)*-"p", E[X] = ;, 1 € {r,r+1,..}, Var(X) = p), %3D • X - Hypergeometric(n, M, N), f(z) = , E[X] = n, Var(X) = N=n(1-). %3D • X ~ Poisson(At), f(z) = A0", z e {0, 1, .}, E[X] = At, Var(X) = At. Common Continuous Distributions • X - Exponential(A), f(z) = de-A, z E [0, 00) E[X] = }, Var(X)= . • X- Erlang(r, A), f(z) = A' , zE (0, 00), E[X] = 5, Var(X) = . Suppose that Duke Energy mu
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