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Unformatted text preview: Exercises ' _ 53 Histogram of 1.000 Gaussian Random Numbers from Matlab
'_"'—1—"—‘_r"'—“"T"‘"'""""—"—"""'I"'——_"l"—" 250 200 '150 100 so , . .
a . i. 3 _ —2 _ .i o _1 2 3 _4 Figure 1.13 Histogram of 1,000 Gatissian random samples. ' EXERCISES ELI Specify the sample spaces appropriate for the following random experiments:
at The weight of a randomly selected student in your probability class
b. The lifetime (waiting time in days) to failure of a computer hard disk.
Your lifetime (in'years).  4 ‘ I 
Two consecutive characters appearing in a newspaper editoxialm
The instantaneous voltage at a 120VAC electric receptacle, measured at a randomly _ chosen time. .
ii. The position (to the nearest minute) of the minute hand on a clock face, observed at 'a randomly chosen time. '
g” The number of alpha particles emitted by a micmgiam of indium in I second.
h The numbeitof photons emitted in 1 millisecond by an LED operating at 1 milliwam
in. The number of green pixels on the ﬁst scan line of a Computer monitor opeiating
at 640 x 480 resolution“ ' '_ ‘ '  ' ~ 131.2 ' Let :2 = Nit,“ Given the two subsets of $2, 7
A = {n : n is divisible by 3} and B = {n :s' is divisible by 4},
specify, by listing the elements, the sets A‘_,A 03, and A03. _ ‘ 999 54 E13 EL4 E]..5 El.6
EL7 EELS EL9
151.10 ElJl Em Chapter 1 Background, Events, and Data LetQ={(x,y):x€R,y'ER};]R2, I I
A=t<x,y) =cher2 51}, andB={(x,y):xy 20L Accurately sketch A, B, C = A n B and D = A U B‘. A radar receiver computes the energies received by its antenna over a ﬁxed time in both
horizontal and vertical polarizations. Let the sample space 9 be the upperhright quadrant
(ﬁrst orthant) of the xyplane—that is, $2 = {(x, y) : x 3 0. y a 0}“ Given the two subsets ofﬂ,
A = {(26.30 :x a 2y} @613 = {(xJ) :xy >1}, sketch the sets A, B,A", B“,A DB, and A UB..
If ' 9=R2, A=={(x,y):x+y 21}. B={(x,y):x50l, then accurately sketch C —. A UB and D =A nB‘
Evaluate the indicator function 134.}; in terms of IA and 13‘.
Using indicator functions, verify that AcB <==> B"cA“. Using the aiithtnetic of‘indicator‘ functions, determine which of the following statements
are correct and which are incorrect:  a, AUB=(AB)UB
bl. (AUB)C=AU(B~—C)
c.. AUBUC=AU(B A)U(C—(AUB)) Using indicator functions, prove the de Morgan law
. r I. . a A n3 '2 (A‘ U 1‘3")"5
Using the inethod of indicator functions, determinewhether
(A  _C') U (B C) =‘Ar'3‘Cc UAcBC‘L a.. Evaluate the indicator function In in terms of the indicator functions 15, [3, and Jo _ if‘D is the event that both A and B, but not C, occur. . .  b. If ID = 1 — [3, then, using Boolean operations, explain the event D in terms of the . events 21,3, and C.‘ _ . I '
c.‘ If 19 = IAIBIC, then, using Boolean operations, explain the event D in terms of the
 eventsA, B, and C'.‘ ' _ ' d.‘ Ipr = 1;, +13, can events A and B co—occur?  a. Express the indicator function In in tennsof the indicator functions IA, 15, and 15
when D is the event that exactly two of the'events A, B, and C occur; bl. Express theindicator function ID ‘in terms of the indicator functions 14, IE, and Ic
when D is the event that exactly one of the events A, B, and C occurs _ (2.. HI}; = (1  IB)(l  1c),'then use Boolean set operations to express D in terms of the Sets A, B, and C.‘
d.‘ If 13 '== 1,; — In, what can you say about the relationship between sets A and B? Exercises ' 151.14 ELIE ' El..16 ELI? Eltls El.19 E120 .... .____._.___..___m..,1'__,.,_______._.___m§§ a. Evaluate theindicator function In in terms of the indicator functions IA, 13, and Ic  if D is the event that A or B, but not C, occurs:
b If 1;; :5 IA, what can you say about therelation between A and D? . c: If 19 a I 3, what can you say about the relation between A and D? _ d.‘ if'A,B, and C partition the sample space 52, what can you conclude about IA +
[B.+.1C? I ' . : The following parts are unrelated to each ether: 3., If L113 is identically‘zero', what must ‘be true of'A and B? (Use our notation for set
operations and relations.) 7  ' b. HA 0 B" = B me, what must be true of'A and'B‘? c.. If' If + I2 E I, what can you conclude about A and B? The following parts are unrelated to each other: a. If' 1,413 is identically Z610, what must be true of'A and B? (Use our" notation for set operations and‘relations)" ' 
b. If [A 13 a 0, what must be true of'A and B? (Use our notation for set operations and relations.)
c. If 1,; — 13 is identically 1, What must be true 'of‘A and B?
(1.. HA 03 m B UA, what must be true ofA and B? '
e. IfA 03‘” = B 0A", what must be true of'A and B? If A, B, and C denote events, then provide expressions, using Boolean setoperations, for
the following events: ' " ' ' a. At least one of them occurs;  b. At most oneof them occur; '
c.. Exactly one of them occurs; d.. None of them occur; e: Only A occurs; f. B occurs; .  
3, At least two occur; h: Both A and B, but not C, occur; i.. At most two of them occur; j. All of them (recur.
Simplify the following expressions: as (AUB)IUA; b: An(AnB);.
c.‘ (AUB)n(AUC); atanayumncy, A computer ﬁle server has three client workstations: W1, W2, W3: At a given time T, we I
are interested in which of theseclients are requesting service.
.a: Construct an appropriate sample space 9 containing eight points, where a given ' point correSponds to a possible conﬁguration of requests?“ ' . .
bi. Identify the events Arc, for k =0, 1,2, 3, that exactly k clients are requesting service. c. Construct the smallest event algebra A containing each of these events Ag: _
no = {a,b._c,a,e, r} andA : {a},B = {1:}, must c = {a,b} or‘D = {a} be in the
smallest algebra .A of subsets of $2 that contains both A and B? If 52 = {a,b, c,d, e, f} provide an example of an algebra A of subsets of 52 that contains {a , b}: 131.22 151.23 E134 ELZS EL26 131.27 31.28 Chapter 1 Background, Events. and Data a. Given a countany inﬁnite sequence of sets {An }, provide an expression for the event
F that only ﬁnitely manyevents of these events occur, also denoted by A” ‘ f .o
it. Repeat part (a) for the event I that infinitely many of these events occur, also
denoted by A" in  or by lim supAn. _ 
Select a sample of English text from newspaper articles.l (a) Record the binary sequence
of vowels (a, e, i , 0,11) and consonants in initial letters of" 100 words; that is, replace
each vowel occurrence by a 1 and each consonant by a 0.. (b) lnformally, how predictable
is the occurrence of a vowel V or consonant _C from your: data sequence? Can you think
of a “simple” formula that predicts the location of the next vowel given the locations of
the preceding vowels? (c) Evaluate the sample mean, median, and standard deviation for
the resulting binaryvalued sequence. (d) Plotlthe fraction of times vowels occur in the
ﬁrst n letters vs. n, 'for n a' multiple of 5.. I I I ' _
We have data {xi} = {4, 5, 2, 0, 1, 4, 2, 1}. Evaluate the sample mean f and the sample
median 12‘. y .
We have data {xi} consisting of the following six numbers: 0, —40, 2, —I, —5, 6..  Evaluate the sample mean 3:“ and the sample median E. . . _  We have data {xi} = {~2, 0, 1, 4, —2, 5}.. Evaluate the sample mean .‘f and the sample
median E. ' r .  _ We havedata {xi} consisting of the following four numbers: 0..43, 1..66, 0.12, 0.29... . Evaluate the sample mean 3', the sample median 32, and the sample standard deviation 6".
I a. Run the command iosmt d.1 on a Unixbased clientserver system and collect 100
seconds of data {p1,m....,p1m} on the number of packets sent by the server each
second 3 = ' . t. .,
b. Using a mathematics program such as Matlab, compute the sample mean and median
and'a measure of the ﬂuctuation (sample standard deviation).
c. Compute the histogram by using I; :' 10 bins. " ' " ’
d. Examine a scatter plot of 19, vs.. pin] to explore predictability. Can you think of" a
“simple” formula to predict p. from p1,.....,p,..1? ' You can download several interesting data ﬁles from the website
http://lib.stat.cmu.eduldatasets. ' ' '  ‘ 
Select, say, the Dow Jones data and Calculate the sample mean and median for 1980
and the sample standard deviation. Calculate the histogram of. values, using b = 20..
Explore the predictability of stock prices by examining a scatterplot of tomorrow’s . price vs. today’s price.  ...
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 Spring '05
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