STAT 2122 Homework # 3 Solutions Fall 2018 Instr. Sonin

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Griffin Spencer
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1 STAT 2122 Homeork Solutions Fall 2018 Instr. Sonin Due Friday, October 5 NAME ( points) Sho all ork on problems! (4) 1. In the... Lotto, you may pick six different numbers from the set Ö"ß ß $ß ÞÞÞß 48 Þ If all six of your numbers are randomly selected from 48, you share in a grand prize pool. If they dra five (four) of your six numbers, you share in the second (third) prize pool. Sol: There are 6 lucky (for you) numbers in a box, and % 2 unlucky. Six are selected /o replacement - hypergeometric dis-n. Sample space = {6-combinations out of 48}, Rœ 48 ' œ"ß 2 271ß512. " a) What is the probability that you ill share the first prize pool? œ R 8Þ 149 "! 8 or T ÐÞÞÞÑ œ T ÐF ÑT ÐF lf ÑÞÞÞÞT ÐF lf F ÞÞÞF Ñ œ ÞÞÞ œ à " " ' " b) What is the probability that you ill share at least one prize? ' & " " 5 48 % 7 % R " œ ' % 2 ' % " ÎR 2,168 ÎR œ R & " % R 1Þ 07 "!. (6) 2. One percent of all individuals in a certain population are carriers of a particular disease. A diagnostic test for this decease has a 97 % detection rate for carriers and a 4% detection rate for noncarriers. Suppose the test for this disease is applied independently to to different blood samples from the same randomly selected individual. Hint: use Notation: E" œ {no disease}, E œ { disease}, F" œ {" st test is positive}, F œ {2nd test is positive}. 2 T ÐE Ñ œ Þ**à T ÐE Ñ œ Þ!"à T ÐFlE Ñ œ Þ!%à T ÐFlE Ñ œ Þ* 7à " " a) What is the probability that at least one test is positive? TÐF F Ñ " Sol: There are to choices ho to use the total pr-ty formula using partition WœE E À " either for the event F F or for its complement G. The ansers ill be of course the same. "

2 With G e have. T ÐF F Ñ œ " T ÐF F Ñ œ " T ÐGÑà G œ Ö both tests are negative " T ÐGÑ œ T ÐE ÑT ÐGlE Ñ T ÐE ÑT ÐGlE Ñà " " " 2 T ÐGlE Ñ œ T ÐF le ÑT ÐF le Ñ œ ÒT ÐF le ÑÓ ß " " b/c of indep. of to testsþ Then T ÐGÑ œ ÐÞ**ÑÐÞ*'Ñ ÐÞ!"ÑÐÞ0Ñ Þ9124 à T ÐF" F Ñ œ Þ!876. b) What is the probability that the selected individual is a carrier if at least one test is positive? T ÐF F le Ñ œ " T ÐGlE Ñ œ " (.0) œ Þ***" " " TElF ( FÑœ œ " TÐE ÑTÐF F le Ñ ÐÞ!"ÑÐÞ***"Ñ TÐF FÑ " Þ0876.""% ('). A system comprised of five independent "".. components. The probability of failure for each ð ð component is given. (The system is orking if there is a path of orking components from + to,). + ð 2.2, ð ð Use our standard notation E, E ß œ "ß ÞÞÞß 5. a) Find the probability that the system orks. Hint: Note that the events E and E form a partition, and use the formula of total probability, 2 T ÐEÑ œ T ÐE ÑT ÐEl E Ñ T ÐE ÑT ÐEl E Ñ. Make to figures representing the system for these to cases to find TÐElE Ñand TÐElE Ñ. if 2 orks then our system has a form "." $. $ and hence T ÐE l E2 Ñ œ T ÐF" F Ñß here ð ð l F" œe" EßF % œe$ E& ; + l, and T ÐF" Ñ œ T Ð E" Ñ T ÐE% Ñ T ÐE" E% Ñ œ ð l ð œ Þ* Þ' ÐÞ*ÑÐÞ'Ñ œ Þ*' b/c E and E are indep. " 4 %.% 5.5 similarly T ÐF Ñ œ Þ( Þ& ÐÞ(ÑÐÞ5 Ñ œ Þ)ß and T ÐE l E Ñ œ ÐÞ*'ÑÐÞ)&Ñ œ Þ)"';

3 if does not ork then our system has a form "". $. $ ð ð +, and hence TÐElE Ñ œ TÐC" C Ñß here ð ð %.% 5.5 G œ E E ßG œ E E ß " " $ % & Hence T ÐElE Ñ œ T ÐC ") T ÐC Ñ T ÐC" C Ñ œ ÐÞ*ÑÐÞ(Ñ ÐÞ'ÑÐÞ&Ñ ÐÞ*ÑÐÞ(ÑÐÞ'ÑÐÞ&Ñ œ Þ(%" Finally T ÐEÑ œ (.)) Ð Þ)"'Ñ ÐÞÑÐÞ(% 1Ñ œ Þ)!" b) Find the probability that component orks if the system orks. TÐE leñ œ œ œ TÐE E Ñ TÐE ÑTÐElE Ñ (.))(.)" 6) TÐEÑ TÐEÑ Þ)!" Þ) 15 Þ) c)* Find the probability that component " orks if the system orks. TÐE E Ñ " TÐEÑ " " $ & " $ % & TÐE leñœ " ; E E œ [ E E ÐE E Ñ] [ E E ÐE ÐE E ÑÑ] œ œ ÒÞÞÞÓ [ ÞÞÞÓß the union of to disjoint events, so T ÐE E" Ñ œ ÐÞ*ÑÒ ( Þ) )(.7 Þ5 ÐÞ7ÑÐÞ5ÑÑ ÐÞÑÐÞ7 ÐÞ5ÑÐÞ6Ñ ÐÞ7ÑÐÞ5ÑÐÞ6ÑÑÓ œ ÐÞ9ÑÒ Þ') Þ" 5)Ó œ Þ7542Þ Then T ÐE1lEÑ œ Þ)!" Þ942.9, or using the diagram for the case T ÐElE Ñ e have T ÐE E Ñ œ T ÐE Ñ T ÐElE Ñ œ ÐÞ*ÑÐÞ)$)Ñ œ Þ(&% Þ(542 " " " " à T ÐElE Ñ œ T ÐE ÐÐE E Ñ E ÑÑ œ ÐÞ(Ñ ÐÞ&ÑÐÞ' Þ) ÐÞ'ÑÐÞ)ÑÑ ÐÞ(ÑÐÞ&ÑÐÞ*Ñ œ Þ)$)Þ " % & $ (4) 4. A family decides to have children until it has four children of the same gender. Assuming TÐFÑ œ TÐKÑ œ Þ5 ß ( : œ ; œ Þ5), a) hat is the pmf of \œthe number of children in the family? B % % $ $ :B ( ) : ; 4 Ð:; :;Ñ 10Ð:; :;Ñ!:; % ) "! "! $ % % :Ð4 Ñ œ TÐKKKKÑ TÐFFFFÑ œ : ; à

4 :Ð&Ñ œ T Ð girls, one boy among first four children, last girl) TÐ boys among first four children, last boy) œ Ð: ; :; Ñß :Ð6Ñ œ T Ð girls and 2 boys among first four children, last girl) T Ð2 girls and boys among first four children, last boy) œ Ð:; :;Ñœ10 Ð:; :;Ñ. Similarly :Ð7Ñœ $ $ Ð: ; : ; Ñœ20 Ð: ; : ; Ñœ!: ;. " 4 " ) "! "! 2 ) For :œ;œ e obtain œ,, ß. b) hat is the expected number of children in the family? I\ œ ")' B:ÐBÑ œ $ œ 5Þ)"5. % % $ $ c)* I\œ B:ÐBÑœ%Ð: ; Ñ!Ð: ; :; Ñ ' 0Ð: ; : ; Ñ (!: ; Þ This expression can be simplified using the equalities " œð: ;Ñ œ: ; :;à $ $ $ $ $ "œð: ;Ñ œ : ; $Ð: ; :; Ñœ: ; $:;ßetc. (5) 5. A small shop orders copies of a certain magazine each eek. Let \œdemand for the magazine ith pmf B :B ( ) The shop oner pays $ 1.50 for each copy of the magazine and the price to customers is $ If magazines left at the end of the eek have no salvage value, is it better to order one, to, three or four copies of the magazine? Hint: Introduce random variables: ] 5 œ the number of magazines sold if 5 magazines are ordered, V5 œ the net profit if 5 magazines are ordered ß 5 œ 1, 2, $ß % and find their probability distributions and expected values. We have.5 2.5, and Sol-n: V œ " 5 ] IV œ 5 I] Þ

5 Then IV" œ "Þ5 2.5 œ "Þ Using the dis-n for the demand \ e obtain the dis-n for ] 2 B 1 2 :ÐBÑ l.2 (1.2) ß then I] œ "Þ8 ß and IV œ I] 2 œ Þ5Ð"Þ8Ñ œ ".5Þ Similarly ] $ B l " ; I] $ œ 2. ß IV$ œ (2.Ñ œ 125 Þ à :ÐBÑ l.2. Ð".5Ñ ]% B l " % ; I] œ 2. %ß IV œ 6 2.5(2. %Ñ œ :ÐBÑ l %!Þ The most profitable is to order magazines. This problem is a simplified version of a ellknon model in operations research and applied economics used to determine optimal inventory levels. This model is knon as the Nesvendor Problem or Nesboy Problem. ( 5) 6 **. The game similar to a game described in Problem 1 as popular in Russia (then Soviet Union) in the 70's and 80's. A participant marked six entries in a table ' * here all 54 numbers ere listed sequentially. After many rounds one statistician noticed that participants from Mosco, Leningrad (no St. Petersburg) and Novosibirsk (three cities ith the most educated population) had higher proportion of large payoffs than participants from other regions in proportion to the number of tickets sold. Ho it may happened? Sol-n: The chances to in are the same for any ticket holder, but the chances to share the prize depend on ho you filled the ticket. The numbers have exactly the same chance of inning as any other set of six numbers. The only thing is, if you pick your on numbers, and you base them on something logical, artificial or else, you are more likely to have to split the jackpot prize because others may be using the same numbers. The most commonly played number combination is If you on ith that number, you ould share the jackpot ith many, many other players. You can find an interesting information about this topic in the paper: Cox, S. J., Daniell, G. J. and Nicole, D. A. (1998) À Using Maximum Entropy to Double One's Expected Winnings in the UK National LotteryßÐsee a pdf file on my ebsiteñþ

Notes on the Unique Extension Theorem. Recall that we were interested in defining a general measure of a size of a set on Ò!ß "Ó.

1. More on measures: Notes on the Unique Extension Theorem Recall that we were interested in defining a general measure of a size of a set on Ò!ß "Ó. Defined this measure T. Defined TÐ ( +ß, ) Ñ œ, +Þ