CS433: Simulation and Modeling Modeling and Probability Review


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1 CS433: Smulaton and Modelng Modelng and Probablty Revew Exercse 1. (Probablty of Smple Events) Exercse 1.1 The owner of a camera shop receves a shpment of fve cameras from a camera manufacturer. Unknown to the owner, two of these cameras are defectve. Suppose that the owner selects two of the fve cameras at random and tests them for operablty. a. Descrbe the experment. b. Lst the smple events assocated wth ths experment. c. Defne the followng events n terms of the smple events n part b. A: both cameras are defectve. B: nether camera s defectve. C: at least one camera s defectve. D: the frst camera tested s defectve. d. Fnd P(A), P(B), P(C), and P(D) by usng the smple event approach. Exercse 1.2 The probabltes of varous numbers of falures n a mechancal test are as follows: Pr[0 falures] = 0.21, Pr[l falure] = 0.43, Pr[2 falures] = 0.28, Pr[3 falures] =0.08, Pr[more than 3 falures] = 0. (a) Show ths probablty functon as a graph. (a) Sketch a graph of the correspondng cumulatve dstrbuton functon. (b) What s the expected number of falures, that s, the mathematcal expectaton of the number of falures? Exercse 2. (Condtonal Probablty) Suppose that a person who fals an examnaton s allowed to retake the examnaton but cannot take the examnaton more than three tmes. The probablty that a person passes the exam on the frst, second, or thrd tral s 0.7, 0.8, or 0.9, respectvely. 1. What s the probablty that a person takes the exam twce before passng t? 2. What s the probablty that a person takes the exam three tmes before passng t? 3. What s the probablty that a person passes ths exam?
2 Exercse 3. (Baye s Rule) 1. Show that P( A B C ) = P( A B C ) P( B C ) 2. Show that P( A B) = P( A C B) P( C B) 3. A manufacturer of arcondtonng unts purchases 70% of ts thermostats from company A, 20% from company B, and the rest from company C. Past experence shows that.5% of company A s thermostats, 1% of company B s thermostats and 1.5% of company C s thermostats are lkely to be defectve. An arcondtonng unt randomly selected from ths manufacturer s arcondtonng unt randomly selected from ths manufacturer s producton lne was found to have a defectve thermostat. Refer to Example a. Fnd the probablty that company A suppled the defectve thermostat. b. Fnd the probablty that company B suppled the defectve thermostat. Exercse 4. (Dscrete Random Varable) Let x be a dscrete random varable wth a probablty dstrbuton gven as x p(x) 1/9 1/9 4/9 a. Fnd p(1). b. Fnd μ=e(x). c. Fnd σ, the standard devaton of x. Exercse 5. (Dscrete Random Varable) A polce car vsts a gven neghborhood a random number of tmes x per evenng. p(x) s gven by x p(x) Fnd E(x).. Fnd σ² usng both the defnton and the computatonal formulas. Verfy that the results are dentcal.. Calculate the nterval μ±2σ and fnd the probablty that the random varable x les wthn ths nterval. Does ths agree wth the results gven n Tchebysheff s Theorem? v. What s the probablty that the patrol car wll vst the neghborhood at least twce n a gven evenng?
3 Exercse 6. (Dscrete Random Varable) You are gven the followng nformaton. An nsurance company wants to nsure a $80,000 home aganst fre. One n every 100 such homes s lkely to have a fre; 75% of the homes havng fres wll suffer damages amountng to $40,000, whle the remanng 25% wll suffer total loss. Ignorng all other partal losses, what premum should the company charge n order to break even? Exercse 7. (Contnuous Random Varable) Let the contnuous random varable X have the followng PDF. 2 x f x e, x 0 ( x λ λ ) = for some λ > 0. 0 otherwse (a) Verfy that ths does defne a PDF. (b) Fnd E(X). (c) Fnd Var (X). Hnt: ntegraton by parts may come n handy... Exercse 8. (Posson Dstrbuton) Students arrve ndependently at the unversty central lbrary. The number of students arrvng n a gven nterval follows a Posson dstrbuton wth a mean arrval rate of 15 students per hour. 1. What s the dstrbuton of the nterarrval process? 2. What s the probablty that 30 students wll arrve between 1 pm and 3 pm? 3. If no students have arrved by 1:30pm, what s the probablty that 10 wll arrve between 2 pm and 3 pm? Exercse 9. (Bnomal Dstrbuton) In the past hstory of a certan serous dsease, t has been found that about 1/2 of ts vctms recover. a. Fnd the probablty that exactly 4 of the next 15 patents sufferng from ths dsease wll recover. b. Fnd the probablty that at least 4 of the next 15 patents sufferng wth ths dsease wll recover.
4 Exercse 10. (Bnomal Dstrbuton) We observe cars arrval to a parkng and t appears that: the probablty of havng one car enterng the parkng s 0.35 durng 5 mnutes from 8 am to 9 am the probablty of havng one car enterng the parkng s 0.45 durng 5 mnutes from 9 am to 10 am 1. What s the probablty of havng 5 cars enterng the parkng between 8H10 am and 8H45 am. 2. What s the probablty of havng 2 cars enterng the parkng between 8H55 am and 9H10 am. Exercse 11. (Exponental Dstrbuton) The lfetme, n years, of a satellte placed n orbt s gven by: f ( x ) = e 0.4 x 0 0.4, x 0 otherwse (a) What s the probablty that the satellte s stll alve after 7 years? (b) What s the probablty that the satellte des durng the frst three years? (c) What s the probablty that the satellte des between 2 and 5 years from the tme t s placed n orbt? Exercse 12. (ChSquare Dstrbuton) Numbers of people enterng a commercal buldng by each of four entrances are observed. The resultng sample s as follows: Entrance No. of People a) Test the hypothess that all four entrances are used equally. Use the 0.05 level of sgnfcance. b) Entrances 1 and 2 are on a subway level whle 3 and 4 are on ground level. Test the hypothess that subway and groundlevel entrances are used equally often. Use agan the 0.05 level of sgnfcance.
5 Problem. (Foundaton of Posson Process) In ths problem, we propose to analyze the foundaton of the Posson Process. Suppose that the tme axs s dvded nto a large number of small tme segments of wdth Δt. Assume that probablty of a sngle clent arrvng n a tme segment s proportonal to the length of the tme segment Δt, wth a proportonalty constant λ, whch represents the mean arrval rate. We assume that Δt s very small so that the probablty s lower than Compute the followng probabltes P(exactly 1 arrval n [t, t+ Δt]) = P(no arrvals n [t, t+ Δt]) = P(more than 1 arrval n [t, t+ Δt]) = 2. What dstrbuton the above equatons are smlar to? 3. Let us assume that Pn(t) = P(#Arrvals = n at tme t) Let pj(δt) be the probablty of gong from arrvals to j arrvals n a tme nterval of Δt seconds. The state of the system s the number of arrvals. Compute the followng probablty P n (t+ Δt) and express t as a functon of the probablty of exactly one arrval and the probablty of no arrvals n an nterval [t, t+ Δt] as n queston Compute P 0 (t+ Δt) and express t as a functon of the probablty of exactly one arrval and the probablty of no arrvals n an nterval [t, t+ Δt] as n queston Prove that for n 1 ( t ) dpn dt dp0 t dt n 0 λ = λp t + P t = λp t n 1 6. Fnd a soluton for these dfferental equatons and prove that the Posson process s: ( λt ) λt Pn t = e n! 7. Compute the mean and the varance of the Posson Dstrbuton and show that they are equal to n = λ t and σ ² = λ t. 8. Compute the followng probablty P(tme between two consecutve arrvals <=t) and show that the nterarrval tme s exponentally dstrbuted. n
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