Homework 1. 1. Two dice are rolled. Introduce the following events: (1) E : “the sum is odd” (2) F : “at least one number is 1” (3) G : “the sum is 5” List the elementary outcomes in each of the following events: E T F, E S F, F T G, E T F, E ¯ T F T G. For this problem, would you care whether the dice are fair? 2. Two fair dice are rolled. Compute the probability that the number on the first is smaller that the number on the second. 3. Let A, B, C be three events such that P(A) = 0.5, P(B) = 0.6, P(C) = 0.8 (a) Can any two of these events be mutually exclusive? Explain your conclusion. (b) Assuming that the events are independent, compute P(A S B S C). 4. Let A and B be events such that P(A) = 0.7 and P(B) = 0.8. (a) Circle the possible values of P(A T B): 0.3 0.5 0.8 0.9 (b) Circle the possible values of P(A S B): 07 0.8 1 You need to explain each of your conclusions. For example, if you think that P(A T B) can be 0.5, you draw the corresponding Venn diagram, and if you think that P(A S B) cannot be 1, you support your claim with suitable formulas. 5. A student is applying to MBA programs at Harvard, Yale, and MIT. Accordingly, the student prepares three personalized application letters and three addressed envelopes. Unfortunately, after three nights of heavy studying, the student is somewhat disoriented and places the letters in the envelopes at random. What is the probability that at least one letter ended up in the correct envelope? 6. At a certain school, 60% of the students wear neither a ring nor a necklace, 20% wear a ring, 30% wear a necklace. Compute the probability that a randomly selected student wears (a) a ring OR a necklace; (b) a ring AND a necklace. 7. A school offers three language classes: Spanish (S), French (F), and German (G). There are 100 students total, of which 28 take S, 26 take F, 16 take G, 12 take both S and F, 4 take both S and G, 6 take both F and G, and 2 take all three languages. (1) Compute the probability that a randomly selected student (a) is not taking any of the three language classes; (b) takes EXACTLY one of the three language classes. (2) Compute the probability that, of two randomly selected students, at least one takes a language class. 8. 30% of stopped drivers are drunk. Sobriety test is 95% accurate on drunk drives and 80% accurate on sober drivers. A driver is stopped and fails the sobriety test. (a) What is the probability that the driver is drunk? (b) After failing the sobriety test on the first try, the driver somehow gets a re-test and passes it. What is the probability that the driver is not drunk? Assume that the outcomes of the two tests are independent. (c) Now supposed that the driver failed the sobriety test twice in a row. How many more times should the driver fail the test to be 99% sure that the driver is drunk? Again, assume that the outcomes of the test are independent. 9. True or false: if A and B are events such that 0 < p(a)="">< 1="" and="" p(b|a)="P(B|A¯)," then="" a="" and="" b="" are="" independent?="" homework="" 2.="" 1.="" a="" coin="" is="" tossed="" n="" times.="" let="" x="" be="" the="" difference="" between="" the="" number="" of="" heads="" and="" the="" number="" of="" tails.="" find="" the="" possible="" values="" of="" x.="" do="" we="" care="" whether="" the="" coin="" is="" fair="" or="" not?="" 2.="" a="" fair="" coin="" is="" tossed="" n="" times.="" let="" x="" be="" the="" difference="" between="" the="" number="" of="" heads="" and="" the="" number="" of="" tails.="" find="" the="" distribution="" of="" x="" when="" (a)="" n="3;" (b)="" n="4." 3.="" consider="" the="" following="" strategy="" for="" paying="" the="" roulette.="" bet="" $1="" on="" red.="" if="" red="" appears="" (which="" happens="" with="" probability="" 18/38),="" then="" take="" the="" $1="" profit="" and="" stop="" playing="" for="" the="" day.="" if="" red="" does="" not="" appear,="" then="" bet="" additional="" $1="" on="" red="" each="" of="" the="" following="" two="" rounds,="" and="" then="" stop="" playing="" for="" the="" day="" no="" matter="" the="" outcome.="" let="" x="" be="" the="" net="" gain/loss.="" (a)="" find="" the="" distribution="" of="" x;="" (b)="" compute="" p(x=""> 0); (c) Compute the expected value of X; (d) Would you consider this a winning strategy? 1 4. Two fair dice are rolled. Define the following random variables: X, the value of the first die; Y , the sum of the two values; Z, the larger of the two values; V , the smaller of the two values. Find the joint distribution of (a) Z and Y ; (b) X and Y ; (c) Z and V . 5. Consider the function f(x) = ( C(2x - x 2 ) 0 < x="">< 2="" 0="" otherwise.="" (a)="" could="" f="" be="" a="" cumulative="" distribution="" function?="" if="" so,="" find="" c.="" (b)="" could="" f="" be="" a="" probability="" density="" function?="" if="" so,="" find="" c.="" 6.="" the="" joint="" probability="" density="" function="" of="" two="" random="" variables="" x="" and="" y="" fxy="" (x,="" y)="n" cy,="" x2="" +="" y="" 2="1," |x|="1," y="0," 0="" otherwise.="" determine="" the="" value="" of="" c="" and="" then="" compute="" ex,="" ey,="" var(x),="" var(y="" ),="" cov(x,="" y="" ),="" cor(x,="" y="" ),="" fx(x),="" fy="" (y),="" fx|y="" (x|y),="" fy="" |x(y|x),="" p(|x|="">< 1/2|y="1/2)." 7.="" the="" joint="" probability="" density="" function="" of="" two="" random="" variables="" x="" and="" y="" is="" fxy="" (x,="" y)="c(y" 2="" -="" x="" 2="" )e="" -y="" ,="" -y="x" =="" y,="" 0="">< y="">< +8.="" find="" (a)="" the="" value="" of="" c;="" (b)="" the="" marginal="" densities="" of="" x="" and="" y="" ;="" (c)="" expected="" value="" of="" x.="" 8.="" given="" the="" joint="" density="" fxy="fXY" (x,="" y)="" of="" two="" random="" variables="" x,="" y="" ,="" decide="" whether="" the="" random="" variables="" are="" independent:="" (a)="" fxy="" (x,="" y)="(" xe-(x+y)="" ,="" x=""> 0, y > 0; 0, otherwise (b) fXY (x, y) = ( 2, 0 < x="">< y,="" 0="">< y="">< 1;="" 0,="" otherwise="" 9.="" five="" men="" and="" five="" women="" are="" ranked="" according="" to="" their="" performance="" on="" a="" test.="" assume="" that="" there="" are="" no="" ties="" (that="" is,="" no="" two="" people="" can="" have="" the="" same="" ranking)="" and="" that="" all="" possible="" ranking="" are="" equally="" likely.="" let="" x="" be="" the="" ranking="" of="" the="" top="" woman.="" find="" the="" distribution="" of="" x.="" homework="" 3.="" 1.="" a="" “traditional”="" three-digit="" telephone="" area="" code="" is="" constructed="" as="" follows.="" the="" first="" digit="" is="" from="" the="" set="" {2,="" 3,="" 4,="" 5,="" 6,="" 7,="" 8,="" 9},="" the="" second="" is="" either="" 0="" or="" 1,="" the="" last="" is="" from="" the="" set="" {1,="" 2,="" 3,="" 4,="" 5,="" 6,="" 7,="" 8,="" 9}.="" (a)="" how="" many="" area="" codes="" like="" this="" are="" possible?="" (b)="" how="" many="" such="" area="" codes="" start="" with="" 5?="" 2.="" in="" how="" many="" ways="" can="" three="" novels,="" two="" mathematics="" books="" and="" one="" chemistry="" book="" be="" arranged="" on="" a="" shelf="" if="" (a)="" any="" arrangement="" is="" allowed;="" (b)="" math="" books="" must="" be="" together="" and="" the="" novels="" must="" be="" together;="" (c)="" only="" the="" novels="" must="" be="" together.="" 3.="" seven="" different="" gifts="" are="" distributed="" among="" 10="" children="" so="" that="" no="" child="" gets="" more="" than="" one="" gift.="" how="" many="" different="" outcomes="" are="" possible.="" 4.="" in="" a="" certain="" jurisdiction,="" it="" takes="" at="" least="" 9="" votes="" of="" a="" 12-member="" jury="" to="" get="" a="" conviction.="" assume="" that="" (1)="" 65%="" of="" all="" defendants="" are="" guilty;="" (2)="" the="" probability="" that="" a="" juror="" will="" convict="" an="" innocent="" is="" 0.1;="" (3)="" the="" probability="" that="" a="" juror="" will="" acquit="" a="" guilty="" is="" 0.2;="" (4)="" each="" juror="" votes="" independently="" of="" the="" rest="" of="" the="" panel;="" compute="" the="" probabilities="" of="" the="" following="" events:="" (a)="" the="" panel="" renders="" a="" correct="" decision;="" (b)="" the="" defendant="" is="" convicted.="" note:="" you="" do="" need="" a="" computer="" to="" evaluate="" the="" probabilities="" numerically.="" 5.="" the="" number="" of="" fire="" alarms="" in="" a="" certain="" city="" has="" poisson="" distribution.="" on="" average,="" there="" are="" 6="" alarms="" every="" 24="" hours.="" compute="" the="" following="" probabilities:="" (a)="" to="" have="" three="" or="" four="" alarms="" in="" 24="" hours.="" (b)="" to="" have="" no="" alarms="" in="" 4="" hours.="" (c)="" to="" have="" more="" than="" one="" alarm="" in="" 36="" hours.="" (d)="" to="" have="" at="" least="" one="" alarm="" in="" 20="" minutes.="" 6.="" (a)="" a="" stick="" is="" broken="" into="" two="" pieces="" at="" random.="" compute="" the="" probability="" that="" the="" ratio="" of="" the="" longer="" part="" to="" the="" shorter="" is="" at="" least="" a,="" where="" a=""> 1. (The length of the stick does not matter; put it equal to 1 if you want). (b) A stick is broken into three pieces at random. What is the probability that the pieces are sides of a triangle? 7. Given a normal random variable X with mean 10 and variance 36, compute the following probabilities: (a) P(X > 5); (b) P(4 < x="">< 16);="" (c)="" p(x="">< 8);="" (d)="" p(x="">< 20);="" (e)="" p(x=""> 16). Please use a table of the standard normal distribution. 8. Compute the variance of the normal random variable X if E(X) = 5 and P(X > 9) = 0.2. Please use a table of the standard normal distribution. 9. An urn contains four black and four white balls. Four balls are taken out of the urn. If two are black and two are white, the experiment ends. Otherwise, the balls are returned to the urn and the experiment is repeated. Denote by X the number of experiments conducted. Find the probability distribution of X. (Note: the probability of success is 18/35; start by verifying this). Homework 4. 1. Let X be binomial random variable with parameters n = 100 and p = 0.65. Use normal approximation with the continuity correction to compute the following probabilities: (a) P(X = 50); (b) P(60 = X = 70); (c) P(X < 75).="" 2.="" let="" x="" be="" the="" number="" of="" cavities="" that="" develop="" in="" a="" 6-month="" period="" in="" the="" mouth="" of="" a="" child="" that="" uses="" the="" new="" brand="" of="" toothpaste="" “cavifree”.="" the="" distribution="" of="" x="" is="" shown="" below.="" c="" 0="" 1="" 2="" 3="" p(x="c)" 0.4="" 0.3="" 0.2="" 0.1="" a)="" a="" family="" has="" three="" children="" and="" they="" all="" use="" cavifree.="" assuming="" that="" the="" number="" of="" cavities="" acquired="" by="" any="" one="" child="" is="" independent="" of="" the="" number="" acquired="" by="" any="" other="" child,="" find="" the="" probability="" that="" between="" them="" they="" acquire="" at="" most="" one="" cavity="" in="" a="" 6-month="" period.="" b)="" find="" the="" expected="" value="" and="" the="" standard="" deviation="" of="" x.="" c)="" a="" boarding="" school="" has="" 150="" students="" and="" they="" all="" use="" cavifree.="" use="" the="" clt="" to="" approximate="" the="" probability="" that="" the="" students="" acquire="" more="" than="" a="" total="" of="" 200="" cavities="" in="" a="" 6-month="" period.="" (again,="" you="" may="" assume="" that="" the="" number="" of="" cavities="" acquired="" by="" the="" different="" students="" are="" independent.)="" 3.="" the="" waiting="" time="" t,="" in="" minutes,="" for="" the="" green="" light="" at="" a="" certain="" intersection="" is="" a="" random="" variable="" with="" the="" following="" probability="" density="" function:="" ft="" (t)="½" 3t="" 2="" ,="" 0="">< t="">< 1,="" 0,="" otherwise="" using="" the="" clt,="" find="" the="" approximate="" probability="" that,="" after="" driving="" through="" the="" intersection="" 60="" times,="" you="" will="" have="" spent="" the="" total="" of="" more="" than="" 45="" minutes="" waiting="" for="" the="" green="" light.="" 4.="" using="" either="" a="" computer="" or="" pencil,="" paper="" and="" your="" knowledge,="" draw="" the="" normal="" probability="" plots="" for="" the="" following="" distributions.="" then="" use="" your="" knowledge="" to="" explain="" why="" the="" graphs="" look="" the="" way="" they="" do.="" f1(x)="1" 2="" v="" p="" e="" -x="" 2/4="" ;="" f2(x)="1" p(x="" 2="" +="" 1);="" f3(x)="cx2" e="" -x="" 2="" ;="" f4(x)="cx4" e="" -x="" ;="" f5(x)="cx-1/2" e="" -x="" 3="" .="" for="" f3,="" f4,="" and="" f5,="" we="" assume="" the="" function="" to="" be="" zero="" for="" x="0" and="" choose="" c="" so="" that="" the="" function="" is="" a="" probability="" density.="" homework="" 5.="" 1.="" the="" lifetime="" of="" a="" toaster="" from="" the="" company="" toaster’s="" choice="" has="" a="" normal="" distribution="" with="" standard="" deviation="" 1.5="" years.="" a="" random="" sample="" of="" 400="" toasters="" was="" drawn="" yielding="" the="" sample="" lifetime="" average="" of="" 6="" years.="" a)="" compute="" a="" 90%="" confidence="" interval="" for="" the="" mean="" lifetime="" of="" the="" toasters.="" b)="" what="" sample="" size="" is="" needed="" to="" find="" the="" mean="" lifetime="" of="" the="" toasters="" to="" within="" plus="" or="" minus="" 0.05="" years="" at="" the="" same="" 90%="" confidence="" level?="" c)="" how="" will="" the="" answers="" in="" parts="" a)="" and="" b)="" change="" if,="" instead="" of="" knowing="" the="" standard="" deviation="" to="" be="" 1.5="" years,="" it="" was="" estimated="" to="" be="" 1.5="" years,="" based="" on="" the="" same="" sample="" of="" 400="" devices.="" d)="" do="" parts="" a)="" and="" b)="" under="" the="" assumption="" that="" the="" lifetime="" has="" normal="" distribution,="" but="" with="" unknown="" standard="" deviation,="" and="" that="" a="" sample="" of="" 10="" devices="" produced="" sample="" lifetime="" average="" of="" 6="" years="" and="" sample="" standard="" deviation="" of="" 1.5="" years.="" e)="" compare="" the="" intervals="" from="" parts="" a)="" and="" d).="" which="" one="" is="" longer?="" does="" it="" make="" sense?="" why?="" f)="" compare="" the="" sample="" sizes="" in="" parts="" b)="" and="" d).="" which="" one="" is="" larger?="" does="" it="" make="" sense?="" why?="" 2.="" in="" a="" survey="" of="" 100="" people="" from="" a="" certain="" city,="" 20="" claimed="" to="" have="" seen="" a="" ufo.="" (a)="" find="" the="" point="" estimate="" of="" the="" proportion="" of="" people="" in="" that="" city="" who="" claim="" to="" have="" seen="" a="" ufo.="" (b)="" construct="" the="" 98%="" confidence="" interval="" for="" the="" proportion="" of="" people="" in="" that="" city="" who="" claim="" to="" have="" seen="" a="" ufo.="" (c)="" how="" many="" more="" people="" in="" that="" city="" must="" be="" surveyed="" to="" estimate="" the="" proportion="" of="" the="" people="" who="" claim="" to="" have="" seen="" a="" ufo="" to="" within="" ±2%="" with="" 98%="" confidence.="" homework="" 6.="" 1.="" a="" weight-loss="" company="" “sleek="" and="" slender”="" claims="" that="" the="" average="" weight="" loss="" of="" its="" customers="" is="" at="" least="" 25="" pounds.="" after="" a="" bad="" experience="" with="" this="" company,="" fred,="" an="" unsatisfied="" customer,="" wants="" to="" perform="" sampling="" in="" order="" to="" reject="" the="" claim="" of="" sleek="" and="" slender="" and="" possibly="" sue="" them.="" he="" decides="" to="" take="" 4="" randomly="" chosen="" customers="" and="" to="" use="" a="" level="" of="" significance="" of="" 5%.="" he="" has="" obtained="" the="" data="" 13="" 10="" 20="" 25="" representing="" their="" weight="" loss="" in="" pounds.="" he="" assumes="" that="" the="" weight-loss="" data="" for="" customers="" are="" normally="" distributed.="" a)="" formulate="" an="" appropriate="" null="" and="" alternative="" hypotheses="" for="" fred="" to="" use.="" b)="" state="" the="" rejection="" rule.="" c)="" compute="" the="" value="" of="" the="" test="" statistic.="" d)="" should="" fred="" reject="" the="" null="" hypothesis="" at="" the="" 5%="" level="" of="" significance?="" explain.="" e)="" what="" can="" you="" say="" about="" the="" p-value="" for="" for="" this="" experiment.="" circle="" one,="" and="" explain.="" (1)="" the="" p-value="" is="" less="" than="" 0.01.="" (2)="" the="" p-value="" is="" between="" 0.01="" and="" 0.025.="" (3)="" the="" p-value="" is="" between="" 0.025="" and="" 0.05.="" (4)="" the="" p-value="" is="" between="" 0.05="" and="" 0.1.="" (5)="" the="" p-value="" is="" greater="" than="" 0.1.="" 2.="" after="" losing="" a="" game="" with="" friends,="" alice="" suspects="" that="" the="" die="" which="" was="" used="" was="" not="" fair.="" she="" suspects="" that="" the="" probability="" of="" “1”="" appearing="" is="" not="" 1/6="" and="" she="" decides="" to="" test="" this="" by="" rolling="" the="" die="" 300="" times="" and="" using="" a="" level="" of="" significance="" of="" 5%.="" a)="" formulate="" an="" appropriate="" null="" and="" alternative="" hypotheses="" for="" alice="" to="" use.="" b)="" state="" the="" rejection="" rule.="" c)="" after="" rolling="" the="" die="" 300="" times,="" she="" noticed="" that="" “1”="" appeared="" 38="" times.="" compute="" the="" value="" of="" the="" test="" statistic="" in="" part="" b).="" d)="" should="" alice="" reject="" the="" null="" hypothesis="" at="" the="" 5%="" level="" of="" significance?="" explain.="" e)="" compute="" the="" p-value.="" 3.="" consider="" the="" problem="" of="" testing="" the="" null="" hypothesis="" a="0" against="" the="" alternative="" a="1," where="" a="" is="" the="" mean="" of="" the="" normal="" population="" with="" standard="" deviation="" s.="" let="" n="" be="" the="" sample="" size="" and="" let="" y="" be="" the="" value="" of="" the="" test="" statistic.="" (a)="" sketch="" the="" graph="" of="" the="" p-value="" as="" the="" function="" of="" (i)="" y="" (ii)="" s="" (iii)="" n="" (b)="" sketch="" the="" graph="" of="" the="" power="" of="" the="" test="" as="" the="" function="" of="" (i)="" y="" (ii)="" s="" (iii)="" n="" 4.="" suppose="" that="" the="" distribution="" of="" the="" test="" statistic="" to="" test="" the="" null="" hypothesis="" a="0" against="" the="" alternative="" a="1/2" is="" f0(x)="2(1" -="" x),="" 0="">< x="">< 1,="" under="" the="" null="" hypothesis="" and="" f1(x)="2x," 0="">< x="">< 1, under the alternative. suppose that the critical region is [c, 1] and the observed value of the test statistic is y, where c and y are numbers between 0 and 1. compute the p-value of the experiment and the power of the test, as functions of y and sketch the corresponding graphs. what do you expect from the power and p-value as y ? 0? as y ? 1? do you expect the functions to be monotone? are you getting the behavior you expect? 1,="" under="" the="" alternative.="" suppose="" that="" the="" critical="" region="" is="" [c,="" 1]="" and="" the="" observed="" value="" of="" the="" test="" statistic="" is="" y,="" where="" c="" and="" y="" are="" numbers="" between="" 0="" and="" 1.="" compute="" the="" p-value="" of="" the="" experiment="" and="" the="" power="" of="" the="" test,="" as="" functions="" of="" y="" and="" sketch="" the="" corresponding="" graphs.="" what="" do="" you="" expect="" from="" the="" power="" and="" p-value="" as="" y="" 0?="" as="" y="" 1?="" do="" you="" expect="" the="" functions="" to="" be="" monotone?="" are="" you="" getting="" the="" behavior="" you="">