Chapter 01 Problems - Exercises ' _ 53 Histogram of 1.000...

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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 120--VAC 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. _ ‘ 99-9 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 xy-plane-—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=(A--B)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, determine-whether (A - _C') U (B C) =‘Ar'3‘Cc U-AcBC‘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 - events-A, 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, occur-s: b If 1;; :5- IA, what can you say about the-relation 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 set-operations, for the following events: ' " ' ' a. At least one of them occurs; - b. At most one-of 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 binary-valued 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 Unix-based client-server 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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Chapter 01 Problems - Exercises ' _ 53 Histogram of 1.000...

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