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midterm2-F06-with solutions

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Unformatted text preview: YOUR NAME: VLADAI PlPldlAs Midterm 2 (Stor 435) November 13, 2006 The midterm has 10 problems. Each problem is worth 3 points. Show \$1 the work and write in the space provided. If more space is needed, write on the back of the pages. Problem 1. Consider the following experiment. In a hat are four slips of paper, with the numbers 1, 2, 3 and 4 on them. These slips are drawn out of the hat one at a time, at random. The outcome is a random permutation of the set {1, 2,3,4}. Let X be the number of slips that are in their “proper places” (for example, X = 2 for the outcome 1432). Make a table of the possible vsluw of X and their probabilities. Solution: 2 3 «5. 4 ~ 3 4 Q 1 Problem 2. A sample-of 2 items is selected at random from a. box containing 10 items of which 3 are defective. Find the expected number of defective items in the sample. Solution: Let X 5: # 017%9MQLMIU‘1 “Mfg Problem 3. On a multiple-choice exam with 3 possible answers for each of the 5 ques- tions, what is the probability that a student would get 4 or more correct answers just by guessing? Solution: X :' 39:93- C-Dmd Mm :- El5, E, ... 3.. ,5. =1:\___.l’h_i " 5.4931 3"“ g5“ .24; Problem 4. Use Poisson approximations to answer the following question. How many people are needed so that the probability that at least one of them has the same birthday as you is greater than 1/2? (Suppose that there are 365 days in a year.) Solution: Soua. ”- If?“ X II- thk'sm Bﬂquzvg ("I ) ' not .4 i :4 lorqml X 8" Roi: (“lg—:5” Kai-Wiley? 32.5 Problem 5. Give two examples of real life distributions which you would expect to be Poisson. Explain your answer in each of the two cases. Solution: For m%— L 94901.19) wtl’h 1:2: h and?“ \:> 2 . Solution: Problem 6. Let X be a discrete random variable with probability mass function P(X = 1) = 1/2, P(X = 2) = 1/4, P(X = 3) = 1/4. Find and draw the cumulative distribution function F of the random variable X. Fial=PlXéal O a<1 J. iéa<2 .2 I. 3, ‘ Rée<3 4 110”}; Problem 7. Let X be a continuous random variable with probability density function _ i, ime'(—1,0), f(\$) = I: if 1' E (09 1): 0, otherwise. Find and draw the cumulative distribution function F of the random variable X. Solution: D _ ' “Lot,{=”~:£l _ Fm: PfYéqi=meab<= Jéé A I "“50 ”—10 Problem 8. A ﬁlling station is supplied with gasoline in the beginning of a week. Suppose its weekly volume of sales X in thousands of gallons is a random variable with probability density function _ 5(1 -z)4, ifs: e (0,1), ﬁx) _ { 0, otherwise. (a) What is the probability that a full tank of 800 gallons will be exhausted by the end of a given week? (6) Compute EX. Solution: {1 4 5__ r 5 [.5 ﬁx 7,3) 5 5 sax) dz = .. than) 533 =(.2) (t) EX: 526 “5-331le =5) fm— 4x+6x"_ 4:: ix Mae "'0 O 1 ‘5 - ‘ L_4 4 sea—”4.3+; 4—1 5-. 35.? :3 -+3--'—- 'L :3. =- 2 3 a“ sx+elo 52-2 3 4 5+6 Q Problem 9. If 65 percent of the population of a large community is in favor of a proposed rise in school taxes, use normal approximations to compute the approximate probability that a random sample of 100 people will contain between 60 and 70 inclusive who are in favor. Solution. X: -l=l' 0i {worm mYQ 51 1.00.. —Bl400’.65) Pleosxorol 5' HEW-“tn XCHJS‘): :mthHH‘i ypmﬂ i Hot. Pm 4 c. Wk? we 4’07?!“ ‘9- 00‘ ‘. 5< X" OU‘_B:<1}DS‘* ‘6 ﬂ _ “.— ams W» mg) PI 2‘" ’ 2am -l.—_- 2. 0.815 --—4 glossy? Problem 10. Let X be an exponential random variable with paramﬂer /\ > 0. Find the number m, called the median of X , for which P(X > m) = 1/2. Show that regardless of A, the median m is his than EX. Solution:' ..>. ' Pun-411:: d:- Plum] .cﬁ’l- lL-Q M) e‘M=3— 0‘ Am=€n9 1—.- u.— ...
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