Practice Exams and Their Solutions Based on

1 Practice Exams and Their SolutionsBased onA Course in Probability and StatisticsCopyrightc 2003 5 by Charles J. StoneDepartment of StatisticsUniversity of California, BerkeleyBerkeley, CA 94720-3860 Please email corrections and other comments to (Chapters 1 6) Practice ExamsFirst Practice First Midterm Exam1. Write an essay on variance and standard LetWhave the exponential distribution with mean 1. Explain howWcanbe used to construct a random variableY=g(W) such thatYis uniformlydistributed on{0,1,2}.3. LetWhave the density functionfgiven byf(w) = 2/w3forw >1 andf(w) = 0 forw 1. SetY= + W, where >0. In terms of and ,determine(a) the distribution function ofY;(b) the density function ofY;(c) the quantiles ofY;(d) the mean ofY;(e) the variance LetYbe a random variable having mean and suppose thatE[(Y )4] this information to determine a good upper bound toP(|Y | 10).

2 5. LetUandVbe independent random variables, each uniformly distributed on[0,1]. SetX=U+VandY=U V. Determine whether or notXandYare LetUandVbe independent random variables, each uniformly distributed on[0,1]. Determine the mean and variance of the random variableY= 3U2 Practice First Midterm Exam7. Consider the task of giving a 15 20 minute review lecture on the role ofdistri-bution functionsin probability theory, which may include illustrative figuresand examples. Write out a complete set of lecture notes that could be usedfor this purpose by yourself or by another student in the LetWhave the density function given byfW(w) = 2wfor 0< w <1 andfW(w) = 0 for other values ofw. SetY=eW.(a) Determine the distribution function and quantiles ofW.(b) Determine the distribution function, density function, and quantiles ofY.

3 (c) Determine the mean and variance ofYdirectly from its density function.(d) Determine the mean and variance ofYdirectly from the density LetW1andW2be independent discrete random variables, each having theprobability function given byf(0) =12,f(1) =13, andf(2) =16. SetY=W1+W2.(a) Determine the mean, variance, and standard deviation ofY.(b) Use Markov s inequality to determine an upper bound toP(Y 3).(c) Use Chebyshev s inequality to determine an upper bound toP(Y 3).(d) Determine the exact value ofP(Y 3).Third Practice First Midterm Exam10. Consider the task of giving a 15 20 minute review lecture on the role ofinde-pendencein that portion of probability theory that is covered in Chapters 1and 2 of the textbook. Write out a complete set of lecture notes that could beused for this purpose by yourself or by another student in the LetW1, W2.

4 Be independent random variables having the common densityfunctionfgiven byf(w) =w 2forw >1 andf(w) = 0 forw 1.(a) Determine the common distribution functionFofW1, W2, ..Given the positive integern, letYn= min(W1, .. , Wn) denote the minimumof the random variablesW1, .. , Wn.(b) Determine the distribution function, density function, andpth quantileofYn.(c) For which values ofndoesYnhave finite mean?(d) For which values ofndoesYnhave finite variance?12. LetW1,W2andW3be independent random variables, each having the uniformdistribution on [0,1].(a) SetY=W1 3W2+ 2W3. Use Chebyshev s inequality to determine anupper bound toP(|Y| 2).(b) Determine the probability function of the random variableY= ind(W1 12)+ ind(W2 13)+ ind(W3 14).Fourth Practice First Midterm Exam13. Consider the following terms: distribution; distribution function; probabilityfunction; density function; random variable .

5 Consider also the task of givinga 20 minute review lecture on the these terms, including Their definitions orother explanations, Their properties, and Their relationships with each other,as covered in Chapter 1 of the textbook and in the corresponding out a complete set of lecture notes that could be used for this purposeby yourself or by another student in the Exams514. LetYbe a random variable having the density functionfgiven byf(y) =y/2for 0< y <2 andf(y) = 0 otherwise.(a) Determine the distribution function ofY.(b) LetUbe uniformly distributed on (0,1). Determine an increasing func-tiongon (0,1) such thatg(U) has the same distribution asY.(c) Determine constantsaandb >0 such that the random variablea+bYhas lower quartile 0 and upper quartile 1.(d) Determine the variance of the random variablea+bY, whereaandbaredetermined by the solution to (c).

6 15. A box has 36 balls, numbered from 1 to 36. A ball is selected at randomfrom the box, so that each ball has probability 1/36 of being selected. LetYdenote the number on the randomly selected ball. LetI1denote the indicatorof the event thatY {1, .. ,12}; letI2denote the indicator of the eventthatY {13, .. ,24}; and letI3denote the indicator of the event thatY {19, .. ,36}.(a) Show that the random variablesI1,I2andI3are NOT independent.(b) Determine the mean and variance ofI1 2I2+ Practice Second Midterm Exam16. Write an essay on multiple linear LetYhave the gamma distribution with shape parameter 2 and scale param-eter . Determine the mean and variance The negative binomial distribution with parameters >0 and (0,1) hasthe probability function on the nonnegative integers given byf(y) = ( +y) ( )y!

7 (1 ) y,y= 0,1,2, ..(a) Determine the mode(s) of the probability function.(b) LetY1andY2be independent random variables having negative binomialdistributions with parameters 1and and 2and , respectively, where 1, 2>0. Show thatY1+Y2has the negative binomial distribution withparameters 1+ 2and .Hint:Consider the power series expansion(1 t) = x=0 ( +x) ( )x!tx,|t|<1,where >0. By equating coefficients in the identity (1 t) 1(1 t) 2=(1 t) ( 1+ 2), we get the new identityy x=0 ( 1+x) ( 1)x! ( 2+y x) ( 2)(y x)!= ( 1+ 2+y) ( 1+ 2)y!,y= 0,1,2, .. ,where 1, 2>0. Use the later identity to get the desired LetWhave the multivariate normal distribution with mean vector andpositive definiten nvariance-covariance matrix .(a) In terms of and , determine the density function ofY= exp(W)(equivalently, the joint density function ofY1.)

8 , Yn, whereYi= exp(Wi)for 1 i n).(b) Let i=E(Wi) denote theith entry of and let ij= cov(Wi, Wj)denote the entry in rowiand columnjof . In terms of these entries,determine the mean and variance 2of the random variableW1+ +Wn.(c) Determine the density function ofY1 Yn= exp(W1+ +Wn) in termsof and .Second Practice Second Midterm Exam20. Consider the task of giving a twenty minute review lecture on the basic proper-ties and role ofthe Poisson distribution and the Poisson processin probabilitytheory. Write out a complete set of lecture notes that could be used for thispurpose by yourself or by another student in the LetW1,W2, andW3be random variables, each of which is greater than 1 withprobability 1, and suppose that these random variables have a joint densityfunction. SetY1=W1,Y2=W1W2, andY3=W1W2W3.

9 Observe that1< Y1< Y2< Y3with probability 1.(a) Determine a formula for the joint density function ofY1,Y2, andY3interms of the joint density function ofW1,W2, andW3.(b) Suppose thatW1,W2, andW3are independent random variables, eachhaving the density function that equalsw 2forw >1 and equals 0otherwise. Determine the joint density function ofY1,Y2, andY3.(c) (Continued) AreY1,Y2, andY3independent (why or why not)?22. (a) LetZ1,Z2, andZ3be uncorrelated random variables, each having vari-ance 1, and setX1=Z1,X2=X1+Z2, andX3=X2+Z3. Determinethe variance-covariance matrix ofX1,X2, andX3.(b) LetW1,W2, andW3be uncorrelated random variables having variances 21, 22, and 23, respectively, and setY2=W1,Y3= Y2+W2, andY1= Y2+ Y3+W3. Determine the variance-covariance matrix ofY1,Y2, andY3.(c) Determine the values of , , 21, 22, and 23in order that the variance-covariance matrices in (a) and (b) Practice Second Midterm Exam23.

10 Consider the task of giving a 15 20 minute review lecture on thegamma distri-butionin that portion of probability theory that is covered in Chapters 3 and4 of the textbook, including normal approximation to the gamma distributionPractice Exams7and the role of the gamma distribution in the treatment of the homogeneousPoisson process on [0, ). Write out a complete set of lecture notes that couldbe used for this purpose by yourself or by another student in the Let the joint distribution ofY1,Y2andY3be multinomial (trinomial) withparametersn= 100, 1=.2, 2=.35 and 3=.45.(a) Justify normal approximation to the distribution ofY1+Y2 Y3.(b) Use normal approximation to determineP(Y3 Y1+Y2).25. LetXandYbe random variables each having finite variance, and supposethatXis not zero with probability one.]