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p195_exercices_5-6

# p195_exercices_5-6 - the probability distribution for Y =...

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Unformatted text preview: '-'- - the probability distribution for Y = it”. " d theprobability distribution for Y = X‘”. Vu'dtheprobabilitydistributionfor Y= lnX. The velocity of a particle in a gas is a random variable w probability distribution M1») = “29—“ V > 0 ubisaconstantthatdependsonthetemperanrreofthe -,. the mass of the particle. 'e-....tl1evalueoftheconstanta. ' .- kinetic energy ofthe particle is W = mV2/2. Find the ' - - : -ility distribution of W. ' Suppose that Xhas the probability distribution fx(x) = 1, 1 5x 5 2 ‘ ' probability distribution of the random variable ('I' 'I'hemndomvariableXhastheprobabilitydism'bmion 0 S x 5 4 so) = g. ‘; probability distribution of Y = (X - 2?. Supplemental Exercises . Show that the following function satisﬁes the proper— ' joint probability mass function: - t/s 1/4 1/4‘ ' M -— er. -- cc? the following: - < 0.5, Y< 1.5) (b) P(Xs 1) < 1.5) (d) P(X> 0.5, Y< 1.5) .7 :4 n I I6 E(X), E(Y), VtX), and KY). .. = probability distribution of the random variableX s-I 't'lonal probability distribution of 1’ given thatX = 1 IX = 1) ” Xand Yindependent? Why orwhynot‘? '. ate the correlationbetweenXand Y. 'Ihe percentage of people given an antirheumatoid .1 -'on who suﬁ‘er severe, moderate, or minor side eﬂ’eets " 20, and 70%, respectively. Assume that people react 5—6 GENERAL FUNCTIONS OF RANDOM'VARIABIES 195 independently and that 20 people are given the medication. Determine the following: (a) The probability that 2, 4, and 14 people will suffersevere, moderate, or minor side effects, respectively (b) The probability that no one will suffer severe side eﬂ‘ects (c) The mean and variance of the number of people that will suffer severe side effects ((1) What is the conditional probability distribution of the number ofpeople who suﬂ’ersevere side eﬁ‘ects givendiat 19 suﬁ‘er minor side effects? (e) What is the conditional mean of the number of people who suffer severe side effects given that 19 suﬂ‘er minor side eﬁ'eets? 5-77. The backoﬁ'torque required to remove bolts in a steel plate is rated as high. moderate, or low. Historically, the prob- ability of a high, moderate, or low rating is 0.6, 0.3, or 0.1, respectively. Suppose thatZDboltsarewaluatedandﬂﬂllthe torque ratings are independent (3) What is the probability that 12, 6, and 2 bolts are ratedas high, moderate. and low, respectively? (b) What is the marginal distribution of the number of bolts rated low? (c) What is the expected number of bolts rated low? (d) What is the probability that the number of bolts rated low is greater than two? (e) What is the conditional distribution of the number of bolts rated low giventhat 16 bolts are rated high? (f) Whatistheconditionalexpectedmmrberofboltsmted low given that 16 bolts are ratedhigh? (g) Arethenumbers ofboltsratedhighandlowindependent random variables? 5—78. To evaluate the technical support from a computer manufactm'er,thenurnberofring‘sbeforeaeallisansweredby asendcerepresentativeistrackedﬂimricelly, 70%afthe callsareansweredmtwonngsorlmm‘mmm threeorfourrings,andtheremainingeaﬁ"" ﬂags ormore.Supposeyoucallthismanufectruur13Imes-Im assmnethatthecallsareindependent. (a) What is the probability that eight calls ave-answered in two ringsorless,onecallisansweredinthreeorfourrings, andonecallrequiresﬁveringsormore? (b) Whatistheprobsbilitythatall lOcallaareansweredin fourringsorless? (c) Whatisﬂieexpectednmnberofcallsansweredinfour ringsorless? (d) Whatistheconditional distribution of the number of calls requiringﬁveringsormoregiventhateightcallsare answeredintworingsorless? (e) What is the conditional expected number of calls requir- ingﬁveﬁngsormozegivenﬂrateightcallsareansweredin tworingsorless? (f) Arethenumberofcallsansweredintworingsorlessand thenan of calls requiring ﬁve rings ormore independ- ent random variables? ...
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