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l (gall/[Ole Prylolem #Ll dnsmg Z/ﬂéﬁi 55W] 1) The true annual return on a $200 investment in Zipuo stock is as follows: Return in percent (X) Return in dollars (Y) Probability 3 (“l l 6 (bl 1 }' 27L a Z; " (It?
3% $6 .4 ' 2 ‘
6% $12 .6 ‘f(°>“taJ % é>(b¢Ia’_12 : Zita
it) Find the mean ofthe random variable X LL 6? variance 2 {£1 and standard deviation ] .41; 7 ; W
b) What are the units? 9 0 (“ii . ‘ . ' . ‘ L _ l . d
5) Find the mean of the random variable Y Ci» b vartance 8 b ILl and standard deviation 2 Cl l  gé / a)Whatarettmunits?EH 'L'a) +602) .469“ 61.53% .12 61—51. tall 2) l he t rue annual retutn on Pizco stock IS identical to Zipeo, and the returns are independent so the true linear conelntion
between them = 0. a) What is the average return (in %) for a lionfolio ot‘SGﬁVo Pizeo and 50% Zipco? 5/ IE “ (ALP) l.§(¢1_r>l= h) Suppose your portfolio is $100 in Pizeo and S too in Zipeo. What is the average return (in 33)? (2‘60 2. '2
~47 (Q 2.15 +51 (2.15) — loft ‘ m
Re) Consider the return i percent'of this portfolio. Find the variance l DES and standard deviation / 034 ' l, 5 <3 2  .  Z .
(at (aw/J M91 2w}: . i?
d) Consider the return in dollars 0 ‘this portfolio. Find the variance 5 lr g g and standard deviation 2 ___Q_75 ("/3 2 e) Would an investor be better off investing $100 in each oftheset two stocks rather titan $2.00 in just one (ignore transactions
costs of buying andfor selling stocks)? Brieﬂy explain your answer. l‘lli‘tlﬁ'i “’7 thlH’l: \l, Nblc  Cacti/N WAIT”, 3) a) Suppose the probability of Bob getting an “A” in any claSS is .2 (20%). Assuming his grades in class are independent of
one another, what is the probability that he takes three classes and receives ALL As? (it ill 1 00? mu 12¢chth w b) What ts the probability that he takes three classes and receives NO As? ._£’.§ P a L .02. 4) You ﬂip a fair coin (probability of head on any one toss) three times: and record whether a head or tail faced up on each of
the three tosses. Each group ofthree tOSSes is one trial or experiment. 68‘} W1 ‘01”. _ —1 he
a) \V1ite1l1e s unple spau.sl1«\\\i11< lll eight ofthe equally likely outcomes: PL 1] M 691‘ “A; LiLliF 11+” (8:: 2) OZ’l1'17lit‘3 M : (T112?
H'U'r FtFT Max:g/QX;3£ [H‘ r FTT ~ 0—2::
llllt r111 b) Let X= number ofheads 1n three tosses. Write the possible values ofX with their associated probabilities: 0 7. 3 @ 2’5"?
L? i? i3 if
4 15° 3’ § 3) What is the true population mean of X? 113;“! — EM” Gall 2(1) 1‘ latU r« is
@hato' 15 the true population variance of X? ox = '3 I 3"“ l ”l
“gb 932 *ECQ’JZ 13—35561 ”sheets?
@Vhat is the true population standard deviation of2 X? ox = @l ’> + "5! 0i)
L , , , 11 at cl 91‘ 6
Gp/1§(23lli)g(8)2#%<85 i‘ 2 ‘ . '§ 1T}: 3Z"§::;— . in 111— P1
3) Suppose you perform the experiment above four times (each single experiment is 3 tosses) and get the following results:
i) HRH; ii) HTT; iii) THH; and iv) THH
3,. l 2, Z. a) What is the sample me in ofX'. ‘1’: :2, 3:1; 2 1 l 1 ,1'. l ') 2 ‘ 12
b) What 18 the sample standard deviation ofX’? SK (1'. (E Q )1 l 5 EEJI: (a) £1) ILL}. 9 . t 7  S
2) Draw a histogram showing your actual outcomes below to the left, with X values on the horizon axis and relativex
frequency ofX on the vertical axis. :1) Draw the true probability histogram (probability density function), below to the right, with X values on the horizontal axis
and the true probability of the corresponding X value on the vertical axis. 1g) '@ i; / histogram for 4 observations true probability histogram 3) (03 Mp 2(3)} .964): !.67L4/\§‘— (,4 x; a ,. {\2 . 2 .2
PHD (h) Q4) ‘ 6.21! x LP'V ; .22 (7:12 71 ‘ 39 (7:12
z wq (m? + ‘69 (a)? : L: m =20 a; = Gaunt/72 >24, C(‘\<T,\: ,7,sz 4,92fy2}Z(AJ.ZG‘P(/“y
‘ “(0on (6’!) [5322‘ 433 (léjéDV‘é‘J ‘
= 4 + H; + 01.6 2M O’plﬁzdzw: WSW 4 {43¢}: am; am CzBOoMpm .
> 2v+/6 :35 0P_:Eb;é. Zi‘)@ MP 9 (e53 L563: 3 HRH (21L) {[ﬂzﬂif +0 “UP—’1? w ¥ QB” b\@1Wml/nm€ncwa 4
:uLlyLLllﬁ iﬁdltf G—qu 2 /155 2
{f 255/)? (120411?! ,1 3'6: (gr 1:ng
= 215331131 N 5%7’L5v‘5‘h; a \hw \ ‘ \\  111111
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z. ILWi. vtl + 4 NZ 
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[Name — LAM aésa'in YtA/ {H/‘IHJKJEMFHE H a) )Go to the Simple Random Sample applet on the publishers web site. Note w entries in the
population hopper as sample size inueases. Does this represent sampling with I  Ithout replacement. y_ YPS ppp I‘PVWlHMS at «Ml six 0’5 557W“? SaraK b) choose a sample of l0. How many of the number In the sample are even? W 04M Mlﬁq. m 4/ aﬁ— :) lt‘your repeatedly took samples of [0, what would be the average number ofeven numbers in a large number ofsamples? LRWNLM 'ﬁ' OW)» P046” f
l8 a) Go to the publisher’s W Roll 4 ' tat is the p0pulati0n mean of sum of spots?
2.9 Its} as f 35 , b) W hat happens to the avera ge ofthe sum of spots as the four die are rolled more and more? .4 9M; Clam la #62: Iltw W/w_ /. )9 Suppose a population has 60% xx omen [Ftwo people are chosen at random.
a) What Is the probability that a woman is chosen first and a man second?_ [6 J [/j 024/ b) That a man ts chosen ﬁrst, and woman second? ; In the sample of 2, one is a woman and the other a man9 _ , . [W :37. E 90) Assume the pIobability of birthdays falling on any one ofthe days ofa standard year is the. same (ignoIe leap years). III a :lass of25 students what Is the probability hat attic/I orgtligre ogfthze Sll}d:2_t5 have tl 133i) birthday Wﬂ/
b 9.: l C”: 5
i'P[A/02 Saw (4Y3 l (”37? 95' 355 H135; [0) Go to the Textbook' 5 website — Statistical Applets — Simple Raﬁ‘d'othSanm “*““‘*“"’—‘ "mg I) Set the population = 100 and sample =10 What is the true proportion of numbers 50 or less?.'
M. 41: {—Jl.h home a sample [1= 10. what is the sample proportion of numbers 50 or less? I
“I 10 et the population— — 12 and sample = 10. What IS thet If
Ih a 33111 1e n= 10. What Is the sam 1e proportion of numbers 6 or less? gloat ill) ;) ) In part b what happen to the probability of choosing a number 6 or less, if more than half of the @ Iumhers are 6 or less? (this Is +§Zifling$ without t‘eplaCement) / I) Go to the Textbook 5 website  Statistical Applets — Probability.
Vhat happens to the sample propel non of tails as [time and mme tosses ale undertaken? appmtthos lop [WPUI‘FMIA \[Ul Ml) ; .5 4) Qumber of strokes that two speciﬁc golfers take on a particular hole are represented by two random variables X and Y
respectively (assume their score on any one particular round is a random variable). X and Y have the following (”true”)
probability distributions: X PROBABILITY Y PROBABILITY
3 .40 3 .30
4 .60 4 .40
5 ' 0 S .30 a) What is the mean nfthe random variable X? Of Y‘7 7
we: Halléﬂ/J: 121741 2.1 ﬂvajwz/etMQ b‘) Suppose two golfers scores are independent of one another. Wha is the mean ofthe sum of the two golfer's scores on this
hole. X + Y, for a randomly selected round? WM zit/11y > 3,6714 . c) Compute the Zstandard deviations ofX ex, and on 0y. What is the standard deviation ofthc sum of the two golfers scores. WY)? (If 21(3 {15? l 6Q 30%: ;14y+.m:,2¢,
01“— 5(% L1 )1 .LC ’4 if); .3[sw( 3 :.6 it? they l (rt1W  ‘i n .21 +1 —. £1165“ uppose a study measures the time for a 91 l merhcal emergency call to get the patien a hospital. Let X = time (minutes)
111' the operator to alen the ambulance service; Y ; the time for the ambulance to reach the patient: and Z = time to bring the
mtient to the hospital. The study ﬁnds that X, Y and Z are independent,
Suppose: ttx = 2.3 ox = 1.6
uy = 25.2 oy =14.4
llz = 10.9 03 = 6.36 1) What is the mean of the total time required, X ’ /
27> Heatwq = @ £6th 06 >) What 15 the VJiiance of the total timeX X‘lY 1Z? '(= 0. K
(towquazéﬁ  24%
Mt J. 901% t 4131453 ’250.°>‘7 ) What is the standard deviation of the total time X+Y+ ‘7 ) Ifthe values 004 Y and Z were positively correlated would the standard deviation of sum X+Y+Z be larger smaller or equal ‘h—
)the answer in QC? WZ.M Qégmsider two independent random variables X and Y X P1X) Y P Y  2_ 2 _ Z
_,_ .73 l. .7 : _Ll' 11 3 3 Z (FY [email protected]‘6J {SQ b—J
‘ 252 +3323);
a) Calculate: mean ofX ”X: r I“ mean on M = , 6 94/
b) Calculate erianc and standard deviations: _ _
(ﬁzz ”lmLﬂcivlJUﬁ‘lda ext: .ﬂlt (7):: ,‘il7 aﬁ= 3‘! 01: .4'7
: lS—gg +_25 2 = ‘ B“ Z .
@—XY 20—ijer _ '9‘,
3) Calculate: mean on Hz: ll 2 variance of = {(38 and standard deviation: G7, _ _i.' go b) Given X and Y are independent, form a table with the values on and their probabilities: Z P Z
~ (2 2 ~ 35 biz) 3.04 ”to . ~i ,L3X7l ‘ IZIEvL/Z
iz=“ C715)"2'
10 = I {Wt7111M @ 3 Consider the returns to stock off m
or 5% its = 8% or: 2% an e and B' with the following mean retum and standard dex‘iat‘ions: o  Nor Fr ml.
Consider a portfolio of40% invested in A an 60% invested in B:
a) What is the average return (in 0/o) for this portfolio? .4 (53+ Mg] 2 2 +4.9 2 59
3) [tithe true correlation between returns on A and B is zero: [DAB :0) then What is the Variance ofthe return m t _5 portfolio?
@ 2 ,. —— ,2 t C od’} , a
3 > {I 1 \ 2
, . __ ‘ I
Eltlz marge = .15 ed + .3493  Me 324/ {3,58 am + 010 ~. .54 + 5.224 = 3 F?
:) What is the standard deviation’.’ E99 l) Il‘the true correlation betwien returns on A and B is zero, pm; : .4, then what is the variance ofthe return on this portfolio? , Alta)
it} T512 453” ,L 2 (1/) 2 3 = [ﬂail 62(9—3.21]+[9[52 : ‘éLan/I :) What is the standard deviation? W ’ ' e ..
l  ') If the true eorrelatioF get new returns on A and B is nilF0, pAB = l, then what is the variance ofthe return on tlmpﬁtfohol—
I“  ‘6” ”3201(2lbo .~ 3‘98”'35 (52%??? W6; 2 L/l13(LJ‘
((37%? 326/ ,) What is the standard deviation? . ‘ (nonfat/o ‘4 6'7; ' Kev I,— a" or an auto insurance company, three outcomes are possible for any policy (measured in thousands of dollars) are: igloss 1:9 “38151 CM @113!) l [04?) 0/) 3r (EN/303 = ’ 4‘4 #:7, 1991/ nail 1054 .05
a toss +1 l9l‘1
.2 5 thousand l a‘) What is the expected net return from a policy on one driver, BtX)? try 2 they have 10 customers what is the expected Q return from all 10 customers? ’imx‘ W/‘(KL /0 Q9): 2,? Wuwayl, ‘2é,bs7
"”t $1 5.083
JUVISPH b 4 ' ' ‘ the outcomes  . .' erent customers are inde endent‘éould we represent the variance and standard deviations by
ixljand ourx ? Yes 0 LS» (circle correct) i134 [l \i  1. “L
H30 0‘1
'ind the variance and the standard a = ' ' fthe return on these 10 customers ._
Gs‘lZQF7ECEEEi) //’&MWWI ‘upposc the probability of Bob getting an “A" in any class is 3 (3 0% . . ssumc he takes
is in class are independent of one another. '. be a random variable for the number ofA’s Bob earnsbui l lenv
the 4 possible X values and the probability associated with each: t( "'k
pé ‘15): (£1) PéVgl e classes per term and his 0 i i ‘2.
~pose Bobs grades for 4 terms are B B B; A B C; A C C; and A A Dr
ire the sample mean and samele standard deviation of for these 4 temrs‘? I? =
a Z 2 A 2 (2 02 , l )\ 2/ .. Z .. 6‘ 6/
‘ ﬁ@~iJ+G—ut6~/H ~ 362' 3Sx'56 ' “Tet
‘_ 7 3 l is) ‘3 I
re the true mean and standard deviation ofX W? l—lx = ox W ‘ 7‘” \L 43] a “ham/)4 gérg’q 37(7)) (2)27) : U mﬁﬂqaﬂﬁaqlu (qL//ﬁ~.q12;(tt’[email protected]—.qﬁ 19276:”;
D +.L{Li(4_37y +‘ .91” ; 5 '278r00LILIH .2797 +ll‘187: [,5 a histogram showing your actual outcomes below to the left, with X values on the horizontal axis and relative
.y of X on the vertical axis. the true probability histogram (probability density function), below to the right, with X values on the horizontal axis
116: probability of the con'esponding X value on the vertical axis
mm for 4 observations true probability histogram ...
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 Spring '07
 Guggenberger

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