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22 Pages

Course: STAT 302, Spring 2011
School: UBC
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Word Count: 977

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model Which does the God use? Consider the experiment when two dice are tossed. If the dice are identiable, the sample space has 36 sample points. If the dice are not-identiable, the sample space has 21 points. If each outcome is equally likely, what is the probability that the outcome is (1, 1)? () January 13, 2011 1 / 17 Which model does the God use? Assuming identiability: the answer is 1/36 = 2.8%;...

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UBC - STAT - 302
Stat 302, Introduction to ProbabilityJiahua ChenJanuary-April 2011Jiahua Chen ()Lecture 3January-April 20111 / 56Probability is a function on subsets of SSuppose we have a random experiment and identied its sample space S . The probability measure
UBC - STAT - 302
Stat 302, Introduction to ProbabilityJiahua ChenJanuary-April 2011Jiahua Chen ()Lecture 4January-April 20111 / 22Independence of two eventsGiven a random experiment and its sample space, there exist many events. We have reasons to believe that som
UBC - STAT - 302
Stat 302, Introduction to ProbabilityJiahua ChenJanuary-April 2011Jiahua Chen ()Lecture 4.2January-April 20111 / 36Conditional ProbabilityMotivation. Updating the probability of an event E to take into account the information that another event F
UBC - STAT - 302
Random VariablesGiven a random experiment and the sample space , we have already discussed how to assign a probability to subsets of the sample space. Yet in many occasions, we are interested in a random quantity that is a function of the outcome, instea
UBC - STAT - 302
ExtraPlease use a cover page to put your name, ID and Section information with your assignment.()January 29, 20111 / 25Poisson Random VariablesSuppose we monitor the occurrence of a centain incident in a xed period of time.earthquakes over 5 in the
UBC - STAT - 302
Stat 302, Introduction to ProbabilityJiahua ChenJanuary-April 2011Jiahua Chen ()Lecture 6January-April 20111 / 41AverageIn a class of students, 30% of them get exactly 60, 50% of them get 80 and 20% of them get 95 in their nal exam. What is the cl
UBC - STAT - 302
Stat 302, Introduction to ProbabilityJiahua ChenJanuary-April 2011Jiahua Chen ()Lecture 7January-April 20111 / 24Buons needleConsider a random experiment in which a stick of unit length is randomly thrown onto a oor painted with parallel lines at
UBC - STAT - 302
Stat 302, Introduction to ProbabilityJiahua ChenJanuary-April 2011Jiahua Chen ()Lecture 8January-April 20111 / 24Example: Uniform distributionIn many applications, the random quantity is equally likely over some closed interval. The corresponding
UBC - STAT - 302
Stat 302, Introduction to ProbabilityJiahua ChenJanuary-April 2011Jiahua Chen ()Lecture 9January-April 20111 / 32Variance of continuous random variablesIf X has continuous distribution with pdf f (x ), then var(X ) = E (X )2 =(x )2 f (x )dxwhere
UBC - STAT - 302
Stat 302, Introduction to ProbabilityJiahua ChenJanuary-April 2011Jiahua Chen ()Lecture 10January-April 20111 / 23Disk exampleConsider a random experiment in which a dart is thrown randomly to disk of radius 1. Assume each point on the disk is equ
UBC - STAT - 302
Stat 302, Introduction to ProbabilityJiahua ChenJanuary-April 2011Jiahua Chen ()Lecture 11January-April 20111 / 21Expectation of g (X , Y )Suppose X and Y are jointly (absolutely) continuous with joint pdf given by f (x , y ). Let g (x , y ) be a
UBC - STAT - 302
Stat 302, Introduction to ProbabilityJiahua ChenJanuary-April 2011Jiahua Chen ()Lecture 12January-April 20111 / 21Sum of two continuous random variablesSuppose X and Y have joint pdf given by f (x , y ). Then the cdf of X + Y is given by x =a y x
UBC - STAT - 302
Stat 302, Introduction to ProbabilityJiahua ChenJanuary-April 2011Jiahua Chen ()Lecture 13January-April 20111 / 24Moments and covariance formulasLet X be a random variable. If X is discrete, E (X ) = x xP (X = x ), where P (X = x ) = p (x ) is the
UBC - STAT - 302
Stat 302, Introduction to ProbabilityJiahua ChenJanuary-April 2011Jiahua Chen ()Lecture 14January-April 20111 / 26Variance formula, againLet X and Y be two random variables. var(aX + bY ) = a2 var(X ) + 2ab cov(X , Y ) + b 2 var(Y ). In particular
UBC - STAT - 302
Stat 302, Introduction to ProbabilityJiahua ChenJanuary-April 2011Jiahua Chen ()Lecture 15January-April 20111 / 14Conditional Distribution; discrete caseLet X and Y be two discrete random variables with joint pmf given by p (x ; y ). In other word
UBC - STAT - 302
Stat 302, Introduction to ProbabilityJiahua ChenJanuary-April 2011Jiahua Chen ()Lecture 16January-April 20111 / 23Conditional distribution: continuous random variablesConsider the case where X and Y have joint density function f (x , y ). Similar
UBC - STAT - 302
Stat 302, Introduction to ProbabilityJiahua ChenJanuary-April 2011Jiahua Chen ()Lecture 3January-April 20111 / 17Example of joint pmfSuppose X and Y have joint pmf given by (after multiplied by 290) x =0 x =1 x =2 x =3 pY ( y ) y = 0 y = 1 y = 2 y
UBC - STAT - 302
Stat 302, Introduction to ProbabilityJiahua ChenJanuary-April 2011Jiahua Chen ()Lecture 18January-April 20111 / 24Conditional means and varianceLet us use the previous example once more. The conditional pmf of X given various values of Y is as fol
UBC - STAT - 302
Stat 302, Introduction to ProbabilityJiahua ChenJanuary-April 2011Jiahua Chen ()Lecture 19January-April 20111 / 24What LLNs do not answerLet X1 , X2 , . . . be a sequence of iid random variables with mean and variance 2 . Denote Xn = n 1 (X + 1 +
UBC - STAT - 302
Stat 302, Introduction to ProbabilityJiahua ChenJanuary-April 2011Jiahua Chen ()Lecture 20January-April 20111 / 16Convergence in distributionLet Xn be a binomial random variable with parameter n and p = /n. It is seen that when n , p 0, while np =
UBC - MATH - 302
Math 302.102 Fall 2010 Assignment #1 This assignment is due at the beginning of class on Wednesday, September 22, 2010.1.Suppose that the sample space S consists of four outcomes, say S = cfw_a, b, c, d.(a) Explicitly list all of the possible events. C
UBC - MATH - 302
Math 302.102 Fall 2010 Assignment #2 This assignment is due at the beginning of class on Wednesday, September 29, 2010.1.A well-known television advertisement claims that there are two scoops of raisins in a package of Kelloggs Raisin Bran. Assume that
UBC - MATH - 302
Math 302.102 Fall 2010 Assignment #3 This assignment is due at the beginning of class on Wednesday, October 6, 2010.1.It is known that in a certain population 5% of all men are colour blind and 0.25% of all women are colour blind. Suppose that one perso
UBC - MATH - 302
Math 302.102 Fall 2010 Assignment #4 This assignment is due at the beginning of class on Friday, October 15, 2010.1.In each of the following cases, compute P cfw_0 &lt; X &lt; 2 where the random variable X has the given probability density function. (a) f (x)
UBC - MATH - 302
Math 302.102 Fall 2010 Assignment #5 This assignment is due at the beginning of class on Wednesday, November 3, 2010.1.One important use of Bayes Rule is in the context of a researcher soliciting responses to sensitive questions ; that is, questions to
UBC - MATH - 302
Math 302.102 Fall 2010 Assignment #6 This assignment is due at the beginning of class on Wednesday, November 10, 2010.1.The exponential distribution has an important property that uniquely characterizes it among continuous distributions, the lack of mem
UBC - MATH - 302
Math 302.102 Fall 2010 Assignment #7 This assignment is due at the beginning of class on Monday, November 22, 2010.1.Let 0 &lt; a &lt; b be given positive constants, and suppose that the random vector (X, Y ) has joint density function c(y x), if a &lt; x &lt; y &lt;
UBC - MATH - 302
Math 302.102 Fall 2010 Assignment #8 This assignment is due at the beginning of class on Monday, November 29, 2010.1.A box contains three white balls and four red balls. Suppose that 100 balls are drawn from this box at random with replacement. Write do
UBC - MATH - 302
Math 302.102 Fall 2010 Solutions to Assignment #11.(a) If S = cfw_a, b, c, d is the sample space consisting of 4 outcomes, then there are 24 = 16 possible events. They can be enumerated by listing all events containing 4 elements, namely cfw_a, b, c, d,
UBC - MATH - 302
Math 302.102 Fall 2010 Solutions to Assignment #21.Let Rj , j = 1, 2, . . . , 50, be the event that the j th box of Raisin Bran contains 2 scoops of raisins. We are told that P cfw_Rj = 0.93 for each j and that the events R1 , R2 , . . . , R50 are inde
UBC - MATH - 302
Math 302.102 Fall 2010 Solutions to Assignment #3 Let A be the event that a randomly selected person is a man so that Ac is the event that a randomly selected person is a woman. Let B be the event that a person is colour blind. We are told that P cfw_B |
UBC - MATH - 302
Math 302.102 Fall 2010 Solutions to Assignment #41.(a) We nd2 2 2P cfw_0 &lt; X &lt; 2 =0f (x) dx =1x2dx = x1 1=11 1 =. 2 2(b) We nd2 2 2P cfw_0 &lt; X &lt; 2 =0f (x) dx =07e7x dx = e7x0= 1 e14 .(c) We nd2 1P cfw_0 &lt; X &lt; 2 =0f (x) dx =0e
UBC - MATH - 302
Math 302.102 Fall 2010 Solutions to Assignment #5 1. (a) Suppose that A is the event A = cfw_John smoked marijuana, and suppose further that B is the event B = cfw_John said yes. The law of total probability implies that P cfw_B = P cfw_B | A P cfw_A + P
UBC - MATH - 302
Math 302.102 Fall 2010 Solutions to Assignment #6 1. The denition of conditional probability implies that P cfw_ X &gt; t + s | X &gt; t = P cfw_X &gt; t + s, X &gt; t P cfw_X &gt; t + s = P cfw_X &gt; t P cfw_X &gt; tsince the only way for both cfw_X &gt; t + s and cfw_X &gt; t t
UBC - MATH - 302
Math 302.102 Fall 2010 Solutions to Assignment #7 1. (a) We must choose the value of c so that b a a y b bfX,Y (x, y ) dx dy = 1. Since dy = (y a)2 (y a)3 dy = 2 6y =b(y x) dx dy = a(y x)2 2x=y x=a a=y =a(b a)3 6we conclude that c= 1. (b) By
UBC - MATH - 302
Math 302.102 Fall 2010 Solutions to Assignment #8 1. Since X has a binomial distribution with parameters n = 100 and p = 4/7, we can use the central limit theorem to approximate P cfw_X 50. That is, we know that if X Bin(n, p), then X np np(1 p) has an ap
McMaster - ANTHRO - 1A03
After reading about how the first European settlers influenced several indigenous tribes of the North and South Americas and how the conquest of the new world altered these native tribes lifestyle and their social structure, I found it appealing to furthe
W. Florida - EEL - 0565
Ford Kirkland (tfk3) Homework #1 1. What, if anything, prints when each of the following C+ statements is executed? If nothing prints, then answer nothing Assume x=2 and y=3. a) cout &lt; x; b) cout &lt; x + x; c) cout &lt; x =; d) cout &lt; x = &lt; x; e) cout &lt; x + y
W. Florida - EEL - 0565
Homework 31. #include &lt;iostream&gt; using namespace std; int main() cfw_ float x; for (x=1; x&gt;0; x+) cfw_ if (x &lt; 0) break; cout &lt; &quot;Enter sales in dollars (-1 to end): &quot;; cin &gt; x; cout &lt; &quot;Salery is: \$&quot; &lt; (200 + x * .09) &lt; &quot;\n&quot;; system (&quot;pause&quot;); return 0;
W. Florida - EEL - 0565
Ford Kirkland C+ Dr. Khabou Homework #2 1. X = 5 y = 2 z = 0 2. a) 8.5 b) 48 c) 39 d) 16.7 e) 9 3. a) false b) true c) false 2. #include &lt;iostream&gt;; using namespace std;int main() cfw_ int a; cout &lt; &quot;Please enter your credit score now.\n&quot;; cin &gt; a;if (a
W. Florida - EEL - 0565
HW #41. #include &lt;iostream&gt; #include &lt;cmath&gt; using namespace std; int main() cfw_ char l = 'y'; double a,b,c; while (l = 'y') cfw_ cout &lt; &quot;Please enter coefficient a: &quot;; cin &gt; a; cout &lt; &quot;\nPlease enter coefficient b: &quot;; cin &gt; b; cout &lt; &quot;\nPlease enter co
Uni. Iceland - HUMANITIES - ÍSE102G
1.tmi Kynningnmskeii Kyn Markmi,nmslsing,bkur Verkefni nafnora greinis lsingaroraKynslenskukk. masculine Karlkyn kvk. feminine Kvenkyn neuter Hvorugkyn hvk. Genderisagrammaticaltermforgrouping nounsintodifferenttypesbasedontheir form. Allnounshavefixe
Uni. Iceland - HUMANITIES - ÍSE102G
2.tmi Sastitmi: dag: Kynnafnora,greinisoglsingarora lognendingar rjrhljreglur Fleirtalanafnora,greinisoglsingarora Samrminafnoraoglsingarora Avxllognendingarkk.et.Lo:heillhreinn No: stll steinn Structure:Vl+l,Vn+n(Vstandsforavowel) Onlyoneconsonantfo
Uni. Iceland - HUMANITIES - ÍSE102G
3.tmi Sastitmi: lognendingar rjrhljreglur Fleirtalanafnora,greinisoglsingarora Samrminafnoraoglsingarora Avxl dag: framumAvxl Frumlag,sagnfylling,andlagReglaumAvxl Stemvowelalteration:a~. Intwokindofcontext:saga 1)Iftheendingcontainsthevowelu.sgu
Uni. Iceland - HUMANITIES - ÍSE102G
4.tmi Sastitmi: dag: Avxl Frumlag,sagnfylling,andlag Fallendingarnafnora,lsingaroraoggreinis nefnifall olfall gufallLo.ogno.kk.et. nf. svangurhestur f. svanganhest gf.svngumhesti blrstll blanstl blumstlheilljakki nf. brnnsteinn f. brnanstein heilanj
Uni. Iceland - HUMANITIES - ÍSE102G
5.tmi Sastitmi: dag: Fallendingarnafnora,lsingaroraoggreinis Fallstjrn PersnufornfnAndlag Maurinnhest Falloreftirsgninnierandlag(andl.)(object) sagnarinnar. Andlageralltafaukafalli: Orar:frumlag+sgn+andlag Andlaggeturhaftlkmerkingarhlutverk (different
Uni. Iceland - HUMANITIES - ÍSE102G
6.tmi Sastitmi: dag: Fallstjrn Persnufornfn Orar Eignarsambnd:Sagnirnareiga,hafa,vera me EignarfornfnOrar Frumlag+sgn+andlag 1)Konanhjlparstelpunni nf.+gf.gf. nf.+gf.gf.Venjulegorar 2)Stelpanhjlparkonunni Andlag+sgn+frumlagfugorar 3)Stelpunnihjl
Uni. Iceland - HUMANITIES - ÍSE102G
7.tmi Sastitmi: Orar Eignarsambnd Eignarfornfn Eignarfall dag: nafnora,lsingaroraoggreinisLo.ogno.kk.et. nf. svangurhundur ef. svangshunds nf. brnnsteinn ef. brnssteins blrkjll blskjls heillvasi heilsvasaLo.ogno.kk.ft. nf. svangirhundar ef. svangr
Uni. Iceland - HUMANITIES - ÍSE102G
8.tmi Sastitmi: dag: Eignarfallnafnora,lsingaroraoggreinis Notkuneignarfalls OrareignarsambndumNotkunef. 1)Inobjectpositionwithcertainverbsand certainprepositions: 2)Toindicatethepossessor(theowner)in possessiveconstructions: Jnsaknarstlkunnar Jnfert
Uni. Iceland - HUMANITIES - ÍSE102G
9.tmi Sastitmi: dag: Notkuneignarfalls Orareignarsambndum kk.no.oglo.nendingarnf.et.No.kk.meiendingunf.et. et. nf.penni f.penna gf.penna ef.penna ft. pennar penna pennum penna Veikbeyging Srstaktbeygingardmi No.kk.meur,l,nendingu nf.et.et.nf.hestu
Uni. Iceland - HUMANITIES - ÍSE102G
10.tmi Sastitmi: dag: kk.no.oglo.nendingarnf.et. Meiraumlo.nendingar(tvkv) Brottfallsrhljsrstofnitvkvrano.og lo.Tvkvlo.kk.nendingar nf.et.kk. kvk. et.nf.viturmaurviturkona gulpeysa et.nf.gulurdiskur ertilbrigiafurendingunni Sama beygingardmi og gulur,
Uni. Iceland - HUMANITIES - ÍSE102G
11. tmi Sasti tmi Brottfall srhljs r stofni tvkvra no. og lo. dag Sagnir Nt sagnaUm sagnir Sgn (ft.: sagnir) = sagnor (so.) Sagnbeyging Mismunandi endingar Innskotsstafur j milli stofns og endinga sumum myndum N srhljavxl: B-vxlNafnhttur Nafnhttu
Uni. Iceland - HUMANITIES - ÍSE102G
12. tmi Sasti tmi Nt sagna dag Meira um nt j-innskotEndingar nt et.1. 2. 3. ft.1. 2. 3. V11 - -r -r -um -i -a V2 -i -ir -ir -um -i -a V32 + S3 - -ur (-r/-/-t)4 f-r, fer-, les-t -ur (-r/-) 4 f-r, fer-, les- -um -i -aAthugasemdir 1) Heldur nh. a et.
Uni. Iceland - HUMANITIES - ÍSE102G
13. tmi Sasti tmi j-innskot dag Srhljavxl stofni: B-vxl Notkun ntarYfirlit1 kalla 2 heyra 3 telja tel- tel-ur tel-ur tel-j-um tel-j-i tel-j-a 4 brjta brt- brt-ur brt-ur 5 f 6 fara 7 lesa les- les-t les- kalla- heyr-i kalla-r heyr-ir kalla-r heyr-ir
Uni. Iceland - HUMANITIES - ÍSE102G
14. tmi Sasti tmi Srhljavxl stofni: B-vxl Notkun ntar dag Framt t sterkra sagnaTjning framtar Engin srstk sagnmynd til a tkna framt (kominn tma). Sagnmyndin nt er notu til a tkna framt Hann fer anga morgun Hn hringir brum aftur au flytja nsta mnui
Uni. Iceland - HUMANITIES - ÍSE102G
15. tmi Sasti tmi Framt t sterkra sagna Myndun Endingar dag Meira um t C-vxl Flokkun sterkra sagna Afturbeygt fornafnC-vxl Srhljavxl t (og lh.t. (past participle) sterkra sagna Kennimyndir sterkra sagna1 2 3 4 nh. t.et. t.ft. lh.t. brjta braut b
Uni. Iceland - HUMANITIES - ÍSE102G
16. tmi Sasti tmi C-vxl Flokkun sterkra sagna Afturbeygt fornafn dag Afturbeygt eignarfornafn BohtturAfturbeyging og eign eignarsambndum eru efn. notu til a tkna eiganda g skoa blai mitt selur blinn inn minn ef eigandi vsar til 1.p.et. inn ef eiga
Uni. Iceland - HUMANITIES - ÍSE102G
17. tmi Sasti tmi Afturbeygt eignarfornafn Bohttur dag t veikra sagnaVeikar og sterkar sagnir Flokkun sem byggist myndun tar Sterkar sagnir t me srhljavxlum (C-vxl) brjta, braut, brutum, broti Veikar sagnir t me viskeyti milli stofns og endingar
Uni. Iceland - HUMANITIES - ÍSE102G
18. tmi Sasti tmi t veikra sagna dag reglulegar veikar sagnir Notkun nokkurra httarsagna (modal verbs)reglulegar veikar sagnir Nokkrar algengar veikar sagnir hafa reglulega beygingu A) Nt eins og einn flokkur veikra sagna, t eins og annar flokkur v
Uni. Iceland - HUMANITIES - ÍSE102G
19. tmi Sasti tmi: reglulegar veikar sagnir Httarsagnir dag: Staaratviksor Samandregnar myndir spurnarsetningum OrarStaaratviksor Atviksor (ao.) beygjast ekki. Staaratviksor tkna: dvl sta (rest at a place), hreyfingu til staar (movement to a place)
Uni. Iceland - HUMANITIES - ÍSE102G
20. tmi Sasti tmi: Staaratviksor Orar dag: Forsetningar AukafallsliirFst fallstring Margar forsetningar (fs.) hafa fasta fallstringu, stra alltaf sama fallinu. f. - um, gegnum, kringum, . eir tala um myndina gf. a, af, fr, hj, nlgt, r, . Stelpan