stat2611e3

# stat2611e3 - STAT 2611 Mathematical Statistics University...

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Unformatted text preview: STAT 2611: Mathematical Statistics, University of Minnesota-Morris I Midterm Exam 3 04/16/2010 ' NAME: K Student ID: Problem I. Let Y have a normal distribution with mean 10 and variance 4. Find P(Y< 6.28) and P(4<Y<l6). (l0 pts) : P( ~3.o<z 49.0) : \_P(2srhv—P(%W3€) = V 2 New 3.0) =}z(0.0l2f)=°ﬂ'22« # 13mm The lifetime of a certain electrical component has an exponential distribution with mean 500 hours. A sample of n=5 identical components is taken. Assuming independence among the components, ~ what is the probability that at least one fails hefore 200 hours? (10 pts) w W” "1/; #[vave bl) = p.53 ' 5?» v \~069%2; qggﬁg , ‘ - =— t 68) V X"’Bm(ﬂ/§, p on? o p r PM?!) I- Plxw) = l- (I)(0.;176&)C l—ameg) / 3 O - Problem 3. The length of time to failure (in hundreds of hours) for a transistor is a random variable Y with -.distn'bution function given by O, y<0 F = ,2 (y) {l—e" , yZO. FindP(Y>110|Y5200).(10pts) POM) 0' 5)“), Pu) , FUJ) MWIW) ’ ' m) ._L'>\ - (I-9‘5'('"€ ) l—€"‘+ -LL' 4/» l e « C’ 2 “V m) 1. 0,189! l, 6,4 0“}?lbmi \$13 Pmlem 4. Let Y be disuibhtcd as f( y) , where f(y) =cy2e""‘.for y> 0:f(y) = 0. yso. Find the value of c that makes f (y) a density function. (10 pts) ' 13M Carma “1 *9 *0” Problem é. , Y 2 ,..., Y“, are independent, identical distributed as Beta(2,l). (a) What is the probability density function of Y , the 6"1 order statistic? (10 pts) (6) MW I “cw W) :- / °/”- WW):- ghwc ‘31., '16 W) (ﬁn—oiﬂ'ﬁﬁ’ﬂ 11"" a” 0'3“)? “:14 i 0/00.. H w Mew-*- 8L liloj‘W‘Wr l3 ‘ (“H”) .1» {>50 8' WWW- \ 4itﬁﬁ ’y‘i‘ELH) £54447 ~ S73 “w ) ’ MD meme 8' was?) .jj-iﬂifilt- " mgr} Hark) ; 8' WW" Jamt~t)ﬂa(t—-- ‘ _ ’ 4 ’ 8% WHY) {6 ia—tfraH: ’ i d ° PM t ’ yo ’2 t— 5?) (alto-5%) W/ Problem 6. We have n=1000 parts; each is defective with probability 0.10 (p). What is the probability that _ 120 or more are defective? (10 pts) \ ~» I “a ( A r a ~1— ( 3 da-X—GC'H e oWé’W/QUE mr f. o 4‘ T“ T’,ZXL NBTM(V\»|60,I) ) an “M Tct N ( 0'0 momma) » :1, 25h”)? m m Awiwte ‘ I to.» ) P t4 vx (an/Ltd“? 437%" “Lay/“0J0 I : 0| PWVT‘E = N? 7Wﬂ2ﬂ’ﬁ7“) LEW t Problem 7. Yl , Y 2 ,..., Y" are independent, identical distributed as N (/1, 0':2 The sampling distn'bution — l " 0' 2 of Y = — z Y, is N [ ,u, ——-J . Prove this by Moment Generating Function Method. (20 pts) 11 H n ...
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