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sampleexam2solution - AMS 310: Survey of Probability and...

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Unformatted text preview: AMS 310: Survey of Probability and Statistics Sampie Test 1. Suppose we have a continumis probability distribution f(x)={c(:£+4i if—4<a:<0 0 otherwise (3.) Find the value of c. (b) Find the cumulative distribution function O - 1| In . @ gém’rmdaé -= C-i%++7%] A)“ = (118446) $361 :> (.1sz 'H’ («B A 5.4+ F1907?) i(( 'Lté’xt'o m 7- % “10‘ t - ,1. 1 +8 Rm: 8 «Hawaii/3: éi’iwfl 4+ ' 8]: 9~ 4'” l --!+ air? 7:;100 S {I it «549 ‘i'x 5 t 2 .— .1 PM}: Jflfirwr-M'Si ‘g ’LH'IX 0 I If 00/0 2. Let X and Y be two discrete random variables with joint mass function C(333—1-2y) ifzr= 1,2,3,y=1,2,3, {3 otherwise 130511.21) :{ (a) Find the value of c. (in) Find the marginal distributions fX and fy(y). {(3) Are X and Y independent? Justify your answer. @ 'Z 5'.— Sftw): 6(5+2)+c(Min-rake)+£(é+gy+c('6+4)-rc(6+e) mm”? ~f-C(‘{+2) +c.(6}+4)+cm+6) -= €06 Fl => 02% €15 @ gimp“??? 341,9);- ‘L $301+”): fiI‘E‘x+a)+¢;(gg+4—)+éfi (hug) an? 3 #(‘i’xtflaflz firi- 7'::.2,5. ’ ,— 7’ ; J, ‘ v ‘ r. {o '5 $Y(1)’d?f% 41%?) 90(3*RJ)+§;(9*19)1~9%(‘1HWe (If—5.9+? eacmsé £1.19) :1: fight)“ i213?) 3. Let X be a continuous random variabie having the probability density function 2 1— : -f0< ,< . f{m)=!( “1 4-1 LG othemise. (a) Find the meanuand variance 0‘2 of X. (‘0) Determined the median Q2, the third quartile Q3, and the 92th percentile of X. -i _ (‘9 J1: $1191”sz = SOFR'XU’CX'NLN = "f; I i Em]: i: Xl~f1X')dti<= S702’xzf(’9t,m2>e c 5 7. ,. 1 2 _ . H I mmw we»? , 27c" " .1 Fm? 5 aurwiw 4? IX i 1 13.- 52;: Fr’xwrx—Ocr’f: => ’X=i“‘—’§* J M _ 7X .@X (9 623 g 5:”): Tfir’xleér' 5’ 797i? . _ in l . I 6mm 97”" “'0‘”??? 7c=m7l7g 4. Let W be a continuous random variabie with exponentiai distribution with parameter A : 2, (a) Find the mean ,u and variance 0'2 of W. (b) Prove that P(W > SIW > 2) = P(W > 3). ~— »— r t. u} (9(a) Determine P(W>é1). guy) _;_ Aw»; Re 1:315) Hm); I", e R Wm) 13 .L z .1. : 6‘ — 9x 4’ . I -19 (B run), I) )’ WW” W71):M ‘ I”F(5’) : 6 meme Pr U - W A, I MW”) PM?” “PM” 6 d!” - 5. The age X of a typical subscriber to a certain newspaper is a. normal random variable with mean [11 = 35.5 years old and standard deviation or = 4.8 years. What is the probabiiity that X is between .30 and 40 years old? h 2 x N Mar-5, 4:8 ) .. -,~ adv—3575' ‘7(aos.><s£ri’) 5 Wit-55"; x?” :P(’i.I45“8 «gas 0.4575) 2 : F('o.é}375)- F(—M+53) S 03 .161} —- 0425! : 0‘70}; 2 6. Let X and Y be two continuous random variables with joint density function _ %{2+$y} ifG<$<1,0<y<i. fag) i {0 otherwise Find P(X > §1Y >51 r in th'WM = WWW 'Y’J” F HYPE) I L i j v I 1 . .. g 7 ,x - i2; * " ~ g i yaw—5:91;. oi - [ P(X>t,Y>§)=ég%(l*10lMdfi 91 i 17 it; 3 y I '2. 7. 5x1 ‘ i! 8 r2- : igt‘gv g glimmd?” (3’)” 3’9) ’0 "I grab??? Mr G i ‘1 g i1 ; fi— PW>5= Mew: WW7 j; 36 A i i J . ._L .._ fiLvfi ; it].— l’w’ill” 1)“ ft m 7!; 7. The mean of a random sampie of size n = 25 is used, to estimate the mean of an infinite population that has standard deviateion cr 2 2.4. Find the probability that the error will be less than 1.2. ’: ><"’r><2*“¥15‘ Em -; fi»:_?:a - Bat, ' y X M '— 5, f Pfiii‘yul MA) A . A [ulfiz é X7“ Li : P('f~1é X’JHHV V be M2? —- M— "5 (1 _. P( :2; a; _ 2.5”) = Fox); (“5612.572 atlas «- 0-0062 0.?37é 8. Let X1, X2, . . . ,ng, be a random sample drawn from a normal population with (unknown) mean ,u and standard deviation 0.94. What is the probability distribution of the sampie mean X? Find the probability that the sample mean X will deviate from the true mean p. by at most 0.02? _ 32 : “""“;:;'*x“" x» m, (Ego P( lit—MM”) : P («mag 5': it,“ £0 0;!) clog. -_‘ £ 0‘0),7 m _ .fi w I P p (we? é 35% " (waif) ' W M f M25) «HM? ...
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sampleexam2solution - AMS 310: Survey of Probability and...

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