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STAT1301 (T5 ans)

# STAT1301 (T5 ans) - THE UNIVERSITY OF HONG KONG DEPARTMENT...

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Unformatted text preview: THE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STAT 1301 PROBABILITYAND STATISTICS I EXAlVIPLE CLASS 5 Discrete Distribution Discrete Uniform Distribution Arandom variable X has a discrete uniform[1, N] distribution if P(X = i) = i, i natural number Bernoulli Distribution ABernoulli random variable X takes on only two values: 0 and 1, with probabilities 1- p and p, respectively. Itspmfisgivenby P(X= i) = p i—(1 p)1, ‘ i= 0,1. Binomial Distribution Suppose that n independent trials, each of which results in a “success” with probability p, are to be performed. If X represents the number of successes that occur in the 11 trials, then X is said to be a binomial random variable with parameters (n, p). Its pmf IS given by P(X— — 1') = (I? p (1— p)"”‘, L i = O, 1, ---,n Poisson Distribution A random variable X that takes on one of the values 0, l, 2 is said to be a Poisson random variable with 1,2,--~,N,whereNisa t parameter A, 11 > 0, ifits pmfis given by P(X =1") = 6—1517, i = 0, 1,... L! Geometric Distribution Consider independent trials, each of which is a success with probability p. If X represents the number of the ﬁrst trial that is a success. Its pmfis given by P(X = i) = p(1 -— p)i"1, i = 1, 2, Negative Binomial Distribution A random variable X which denotes the number of trials needed to amass a total of r successes when each trial is independently a success with probability p, then X is said to be a negative binomial random variable with. parameters p and r. Its pmf is given by P(X: i) = Cr_11pr(1 — [Di—r, i = 1”,?“ + 1, Hypergeometric Distribution If there are 2 types of objects which N are of type I and M are of type II. If a sample of size of n is randomly N chosen, then X, the number of type I selected, has pmf P(X= 1')— —- CC‘TﬁLZIE, max[M- n, O] S i S min[n, N]. Ex. 1 Find E(X) if X 15 a random variable with discrete uniform distribution [.4, B], whereZA and B are natural numbers with A(< B. Hence, show that Var(X) —- @411th 2112M): mx )1 1/ z; gnaw—3.43134 {2(5/4ﬂ ’éﬁld +35 f2]; 24] .1 K‘ﬂf : QE’H )(51L/11514/4 L“3#5+35~3A:y , 574d {265:0 14/ EM‘ 2.1;? ’ E06): org/1‘)? :(M)[5*/4*{2(/5’/4f2) ( (2057414!) ‘ : ﬂ 2 W WAX) r. 50(1), @(XJJZ [7/ 2 (z (5441‘ ﬁéw 3% amen) (5741‘ I U W 14L 65125 W+WBJ Ex.2 Atypesetter, on the average, makes one error in every 500 words typeset. A typical page contains 300 words. What is the probability that there will be no more than two errors in ﬁve pages? at rm «a we (ﬂab/cm ﬁrm) ”(M M’r‘l ﬁmlw):€0(52,) 7r (H (SW—x a; /§17D(#)2_3 , ow .L 4 , L * {idea (970 (165) WKXéL)a¢6 @6633?) 1: 0,¢Z}0 :0‘(€23b A! ’AO>0 A; , A\ 7. o Ex.3 Anewsboy purchases papers at 5 dollars and sells them at 6.5 dollars. However, he is not allowed to A; , A1 )1 a return unsold papers. If the daily demand is a binomial random variable with n = 8 and p = 0.5, ' #4-}, <, 0 approximately how many papers should he purchase so as to maximize his expected proﬁt? [9% (E 566th dined my) [1275/45 ‘65?) l jig, no «Maya/5 Arm :6“ 06V) <é,S‘EC‘Z"),)//< M l 3' g 3 ”a K 54 29% 7g W at mm; (mm) c as [éﬂ/pjpévfﬂefﬁ X r 1 X<t ’ > —-"" Fab” (K ‘7: _ we Xvﬁmﬁﬁﬂj [5/4 we; 29/) L:p/%}é %: 6,5147’SK 2 [,S' ’4)“ ﬂéxgk) swap 6C7): Eat/Wt gut/”09 = 65(913076723—[76ﬂkﬂ * 1‘ 6‘ Z7 :— ﬂms 22:0, [teem/2s ” éQ’WW ﬁfe/4W0? z - MQFOWKS - /(X\$})<e.3632é‘rrzv ”mm F00 zcmetxﬂg—am, Aa—Aaqﬂaxﬂﬂl" Ex.4 An auto parts store has 200 rebuilt starters in stock, of which 4 are defective. In response to a purchaseﬂ‘kﬁ order, they randomly select 3 starters from their stock. What is the probability that the customer is given 2 defective starters? ' Cd: X:— yLO- D’F Mia éfW/JI [email protected](NfM9wDr/\fs4l ”my (a it We» (a? e WUWIW .. Ex.5 Let g(x) be a function with —00 < E(g(X)) < 00 and -—00 < g(—l) < 00. Prove that (a) ifX~Poisson(/l),then E(Ag(X)) = E(Xg(X — 1)). ' (b) if Y~negative binomial (r, p) and X = Y — r, then 13((1 — p)g(x)) = E( X gcx ~ 1)) . ﬂ) Kaﬂﬂjsm [a A] ‘fev/uej Ammo) F(x;7():g2i(f a": 04.2fm {757:3 =C,Z’(‘p"5[,f7)y'”, pagan/72!." :12 CX)J= Eta at ﬁr "Mﬁ‘r’ W“ F0)? 49/ 2 _-' L 5 r— 5 jinx“ . ’PCX=<2F)=CZS /7 QZH’Y‘ rw - =— 55095:? — ﬂ [4300,]: ([1205 MC x ”(/7 K Q“? YEW = g 5 (32 —U 47:37:? ”(270)H .- H - E 2 C CE: 2 at? a”; / rr -r H vi) ~ E- E- r - :79, )5?!) FEZ—( Ci; FK/F) 0" ~() Hz /\ n E : go [lg—F r+:—( Ca {ﬂ (7%)] ...
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