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# midterm2solution - AMS 310 Survey of Probability and...

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Unformatted text preview: AMS 310: Survey of Probability and Statistics MIDTERM II Spring 2009 Last Name: —_____ First Name: __—___ ID: Show all your work for full credit 1. Let X be a continuous random variable having the probability density function _ l—Isc| if—CSISC, f(:1:) _ { 0 otherwise. where c > 0 is a constant. (a) (2pts) Plot the function f (x) E. ‘ “C. ‘3‘. L, (b) (3pts) Find :2. C 7- I (b) (3pts) Find the variance VarLX). Efxl -_:. 0 . c c I, _ t5 1 EDI] 5., S OCIAKXNLX .-_ g -7( (ya- y)M + Sp X (lv’XﬂUeI "I; -—i 2. S .J— ...L - I ‘ ‘ : / z “ *‘ l9. '1' ’7‘ .. Z {farLX} [[3] COS) (d) (4pts) Determine the ﬁrst quartile Q1 and the median 62;: of X. (440“? ii X44 Row! if 701 .1 _, .x .x ., .\ H'X)’ gamma: «19+ 7H4; at ~I«='X<v «I 2. Let X and Y be two continuous r.v.'s with joint density function 3(2+xy) ifogstosygi, f(\$1y)={0 otherwise (a) {5pts) )Find the marginal distribution fy(\$). ”SW“ 39% ‘3?"‘9 if: (amigwlg‘: 5%;- ( ‘§‘+9~) (C ) (Spte) Find the probability P(X;_ < 2,Y g 5). J, PlXéJQJ‘l/ﬁ {29" g? S: \$431+ 7(9):” (1:: i ’95:: H- 31):; (i9 H L81 .4. ﬂ? L—qa (d) (Spts) Find the conditional probability P(X > ﬂ)” > %). i y 9‘3 Sammwira PQY)_L)= P(>(>_L): (.— P‘xéi): '36 ., ,.i W MM“ grumwwa = as err-aw ii“- i :5 14+ id- ”IE; 3. Let W be a continuous random variable having exponential distribution with parameter A = 2.5. "' (a) (4pts) Find the mean ,u and variance 02 of W. . l '- ’ ﬂzaLz—szoslf Q‘Z'Livi:0.(£ (b) (4pts) Find P(W > 6. SEW > 2 5). e t QJ’G 5’3 m nae-mat 54 V‘N’k d “[11" ”(U Paws-5] W Arr) IOU-*4) 3 (- le‘ill"); Q = (2 4. Suppose that X and Y are two discrete r.v.’s with joint mass function _ a(r+y) for a: : 0,1,2 and y : 1,2,3 f(z,y) _ { 0 otherwise 2 Z 75191.3)=| 2) C([H- 2+?) 1* 21-15%; +5+4+5J (a) (513m) Find the value of a. (‘J'l 7‘3 I 6'1 2 ‘2 7 (b) (Spts) Find the marginal distributions fx(\$) and fy-(y ). ix”): ,3 90mg): ELIK+1+9<+2+ 1+5}: 72'7”"? (0) {5pts) Are X and Y independent? Justify your answer. Slnb‘t RON/a) \$ £3190 VYLB) , ‘H/ULDa owe n01" \~V\(L’A—t;u€1ﬂhfi (d) (5pts) Compute Vm"(Y). ' 4 _ - _, 5?» (g 1 '3- ’ ;j Mil-{LT} ’"ﬂ—H H (7 5. Let X be a continuous random variable with cumulative distribution function 3 i 0 ifccSa, F(\$): E ifagmgb, 1 ifIZb. and ietY=X2. a La? 4: 1,, 5}th 6135 b (a) (3pts) Find P(X g 2iii) ‘ b . 2&1 __ 'mm :11 (b) {3pts)1fa=—-3andb:4,ﬁndP(lX|£%). WM”; : P(’J;\$X~<3} ( o I“; XS" .__ #- Fiﬁ): 2 .7373. £39 ~35x<£f " REL. '— {5%) i i-fxw : ,1. _ 3, : J. (c) (4pts) Ifa=0andb=2, ﬁnd Porn/Si). ‘7‘ [If 7 a I 750 ‘ ' - 1.1 j. Fm: ,2: 5; 04¢” PM” E 4) .1 i «8 arm 6. (lUpts) Suppose that the daily temperature in an area during the summer is normally distributed with mean ,u : 72° and standard deviation 0 : 6°. What is the probability that on a particular day the temperature gets above 81°? X '4’ (lift? 7% 3 6 ) P(X? a) - _ “ﬂ; gaze) : P( 2 2,115) =1» my) ‘ 5 6 : l~0~?aia =—0ﬁ563 7. The length of human pregnancies is normally distributed with mean ,u = 266 days and standard deviation 0 = 16 days. (a) (5pts) What percent of pregnancies last between 240 and 280 days? (b) (5pt) Suppose an unusually long pregnancy is the one that is in the top 2%. Determine the length of pregnancy that separates an unusually long pregnancy from on that is not unusually long (the 98th percentile). " ,N 2’ G.) X ‘ M(166 ; l6 ) 24-0-2165 L 2 s 231.0 ZéL) 4 P(;14_05_.X.<.9\873)= P( , - (e . H .ﬁ F's V \ T\ r-\ l \_J \l 0 C242: Q— O (.7\ l C t'." 21 ch. - , wait, «.959 _ ,_ 5 , @ P(Xs~.ﬂ):0ﬁg :) P( din—J’s ElL-L—EL ’0-18 7) (—(ﬂF—w 8. (lOpts) An astronomer wants to measure the distance from his observatory to a distant star However due to some disturbances, any measurement will not yield the exact distance (1. Therefore, he has decided to make a series of measurements and then use their average value as an estimate of the actual distance. If the astronomer believes that the values of successive measurements are independent random variables with mean d light years and a starndard deviation of 2 light years. how may measurements need he make to be at least 95% certain that his estimate is accurate within 0. 5 light years? ”7522 276 h —..—. l? of: WscLi/Qwﬁwi‘s via-i XUW} +9 ﬁshwd'e cl. P-(lYMJ -—d{< 6.5") = 9570 ‘> P('05'¢ )(uq)~d<c5‘)= 42.2. 10.6;5 5? Fb-g)" FFJ?’ u 5 - “-7" 2 __, ix!“ : mate 9. (Bonus) (Spts) Let X and Y be independent random variables with common cumulative distribution function F. Let U = max{X, Y}. Find the cumulative distribution function of U . Emmy P(MEV9: P(mwyh“Yléu) :P(xeu,Yéu) Ptxéu)P(YéM) :; 4 PF“) ...
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midterm2solution - AMS 310 Survey of Probability and...

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