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}.

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).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].

3 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.

4 (b) Determine the distribution function, density function, and quantiles ofY.(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).

5 (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, ..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.

6 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).

7 Fourth Practice First Midterm Exam13. Consider the following terms: distribution; distribution function; probabilityfunction; density function; random variable. 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.

8 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).15. A box has 36 balls, numbered from 1 to 36.

9 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.

10 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!(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!