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Exploring Probability and Statistics: Key Concepts and Problems

School
Stony Brook University**We aren't endorsed by this school
Course
AMS 122
Subject
Statistics
Date
Dec 11, 2024
Pages
4
Uploaded by AmbassadorProtonApe5
Problem Set 4StatisticsFall 20221. Suppose a person infected with coronavirus went to a maskless indoorparty with 11 other people. Of the 11 people, 4 stood 3 feet apart fromthem and talked loudly with them for an hour.The other 7 peoplemaintained a distance of at least six feet throughout the entire party.Suppose the infected person has a 40% chance of infecting people whostand within 3 feet and a 10% chance of infecting people who are inthe same room but maintain a distance of at least 6 feet. LetIbe thenumber of people the person infects at the party. FindE[I]AnswerYou can think ofIas the sum of two binomial random variables.•Lis the number of people who talked loudly who got in infected–N= 4,p= 0.4•Dis the number of people who maintained a safe distance–N= 7,p= 0.1I=L+D→E[I] =E[L+D] =E[L]|{z}4×0.4+E[D]|{z}7×.1= 2.3So we expect that 2.3 people will get infected.2. Suppose three quarters of the people in a city make$40,000/year andone quarter make$80,000. LetXbe the income of a randomly chosenperson from that city.(a) What isE[X]?Answer(3/4)×40,000 + (1/4)×80,000 = 50,0001
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(b) What isV ar(X)?Answer(3/4)×(40,000-50,000)2+(1/4)×(80,000-50,000)2= 300,000,0003. Suppose there is a bag of 100 envelopes.Xenvelopes hav a$1 bill insideand 100-Xenvelopes have nothing inside. Suppose I randomly pullan envelope out of the bag. Let the random variableYbe the amountof money in the envelope you pull out. Write out an expression for thevariance ofYusing the variableX(for example, if the variance ofYwasequal to whateverXis equal to divided by 25, thenV ar(Y) =X/25would be your answer).E[Y] =X/100×1 + (100-X)/100×0 =X/100E[Y2] = (X/100)×12+ (100-X)/100×02=X/100V ar(Y) =X/100-(X/100)24. You roll 3 dice. LetMbe the value of the highest value shown on anyof the three dice (ie if you get 2-4-4 thenM= 4).(a) FindP(2≤M≤4)AnswerIt is helpful to find the CDF ofM. LetD1be the value shown onthe first die,D3the value shown on the third die.FM(m) =P(D1≤m∩D2≤m∩D3≤m)=P(D1≤m)×P(D2≤m)×P(D3≤m)=(m/6)×(m/6)×(m/6) =m3/216We now haveFM(m) =m3/216 and we can use it to findP(2≤M≤4):P(2≤M≤4) =P(1< M≤4) =FM(4)-FM(1) = 43/216-13/216 = 63/216(b) FindP(M= 6)AnswerP(M= 6) =FM(6)-FM(5) = 63/216-53/216 = 91/2162
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(c) FindE[M]AnswerE[M] =1×FM(1) + 2×(FM(2)-FM(1)) + 3×(FM(3)-FM(2))+ 4×(FM(4)-FM(3)) + 5×(FM(5)-FM(4)) + 6×(FM(6)-FM(5))E[M] =FM(1)×(1-2) +FM(2)×(2-3) +FM(3)×(3-4)+FM(4)×(4-5) +FM(5)×(5-6) +FM(6)×6E[M] =FM(6)×6-5Xm=1FM(i)!E[M] = 6-5Xm=1m3/63= 4.96(d) FindV ar(M)AnswerV ar(M) = (1-E[M])2×FM(1) +6Xm=2(m-E[M])2×(FM(m)-FM(m-1))(You can quickly do that arithmetic in R:> exp_val_m = 6 - sum((1:5)^3 / 6^3)> m = 2:6> (1 - exp_val_m)^2 * 1^3 / 6^3 ++sum((m - exp_val_m)^2 * (m^3 - (m-1)^3) / 6^3)[1] 1.308449The variance is 1.39R QuestionLoad the New York City Census data that is available on Brightspace. Usingthe data, find the average income in New York City of the following threegroups:3
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•People without a college degree•People with a college degree in a subject other than economics•People with a college degree who majored in economicsNow re-calculate the average for those three groups, but limit it to peoplewho are under age 30 and have an income above zero.4
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