ࡱ> AC@@ bjbjFF ",,,2222222F$FE$RY2ddd22VVVd22VdVVV@22 wL 0Ex 0 FF2222 2DV FF$ j $@FFj  Statistics 101 Review Example on Bivariate November 8, 2004 A company makes two calculators: a scientific calculator and a business calculator. Let X = Monthly demand (in thousands) for the business calculators. Let Y = Monthly demand (in thousands) for the scientific calculators. We are given the following joint probability distribution function: x 1 2 3 4 1 .2 .1 .1 0 y 2 0 .1 .1 0 3 0 .1 .1 .2 Business Calculators Scientific Calculators Selling price $30 $50 Manufacturing cost $20 $30 Total Fixed cost $50,000 per month Questions a) Find p(x). Use p(x) to find the expected value and variance of X. b) Find p(y). Use p(y) to find the expected value and variance of Y. If we sell 1000 scientific calculators, find the probability distribution, expected value and variance for the number of business calculators sold. Let Z be monthly profit (in $10,000 units) from both types of calculators. a) Express Z in terms of X and Y. b) Find p(z). Use p(z) to find the expected value and variance of Z. c) Find Cov(X,Y) and Corr(X,Y). Interpret the correlation in context. d) Use the rules for E(Z) and V(Z) to verify your answer in part b). a) What is the probability that total monthly profit exceeds $5,000? b) What is the probability that total monthly profit is negative? Solutions 1. x p(x) xp(x) (x-()2p(x) y p(y) yp(y) (y-()2p(y) -- ----- ------ ------------ -- ----- ------- ------------ 1 .2 .2 (2.25) .2=.45 1 .4 .4 (1) .4=.4 2 .3 .6 (.25) .3=.075 2 .2 .4 (0) .2= 0 3 .3 .9 (.25) .3=.075 3 .4 1.2 (1) .4=.4 4 .2 .8 (2.25) .2=.45 --------------------------------------------------------------------------------------------- E(X) = 2.5 V(X) = 1.05 E(Y)=2.0 V(Y) = .8 2. x p(x|y=1) xp(x|y=1) (x)2p(x|y=1) -- ---------- ------------ ------------------- 1 .2/.4 =.5 .5 (1) .5 =.5 2 .1/.4=.25 .5 (4) .25= 1.0 3 .1/.4=.25 .75 (9) .25= 2.25 E(X|y=1) = 1.75 E(X2|y=1) = 3.75 V(X|y=1) = 3.75-(1.75)2 = .6875 a) Z = X +2Y-5 b) Table of values of Z x 1 2 3 4 z p(z) zp(z) (z-()2p(z) ----------------------------------- -- ---- ----- ------------ 1 -2 -1 0 1 -2 .2 -.4 (12.25).2=2.45 y 2 0 1 2 3 -1 .1 -.1 (6.25).1= .625 3 2 3 4 5 0 .1 0 (2.25).1= .225 1 .1 .1 (.25) .1= .025 2 .1 .2 (.25) .1 = .025 3 .1 .3 (2.25) .1 =.225 4 .1 .4 (6.25) .1 = .625 5 .2 1.0 (12.25) .2=2.45 ------ --------------- E(Z)=1.5 V(Z)= 6.65 E(XY)= (1*1).2+(2*1).1+(3*1).1+ (2*2).1 +(3*2) .1 + (2*3) .1 +(3*3) .1 +(4*3) .2 = 5.6 COV(X,Y) =E(XY)-E(X)E(Y)=5.6-2.5(2)=.6 CORR(X,Y)= .6/[1.05*.8]1/2 =.6547 The demand for the two calculators tend to go in the same direction. This is a relatively strong (.65 on a 0 to 1 scale) linear relationship. E(Z) = E(X+2Y-5)=E(X) +2E(Y) 5 = 2.5 +2(2) 5 = 1.5 V(Z) =V(X+2Y-5)= V(X) +4V(Y)+2(1)(2) COV(X,Y) = 1.05+3.2+4(.6)= 6.65 a) Profit is greater than .5 ($5,000). Just add up prob. of cases = .1 +.1 +.1+.1+.2 = .6 b) Profit is negative only when x=1 and y=1 or x=2 and y=1 Prob=.2+.1=.3  <BCDw V )*+,TVWXYZ];<H<lmnosMüôðüèèðǰh#uh#uH* jmh#uh#uh7,h7,H*h7,h[=$H*h7,h[=$h[=$H* jmh[=$h[=$h?Ohh?O5>*hh5>*CJaJh+]5>*CJaJhh?O5>*CJaJh7CD# $ h i ! R S < v w x C h^hgd?O & Fgd?O  !gd [gdC D % G ;<_^gd & Fgdgd?O^gd?O & Fgd?Oh^hgd?O_0<  H]tN3'h^hgd7,^gd#u & Fgd7,h^hgd?O23Yyz{}'01?@JMO8_`a|h7Phr{ohq`|hq`|H*h+]hq`|hOh#uh7,h?O-' }~`a^gd7P & Fgd7P^gdO^gdq`| & Fgdq`|h^hgd7, 1h/ =!"#$%@@@ NormalCJ_HaJmH sH tH DA@D Default Paragraph FontRi@R  Table Normal4 l4a (k@(No List,CD#$hi!RS<vwxCD%G;<_0< H  ]  t N 3' }~`a00000000000000000000000 000 000 000000 00 00(00 000 0 0 0000x00x00000000000000 0000000000000000000x 0000x 000 00< H  t ^>00N\>00\>00*7\>00^>00N\>00\>00\>00\>00^>00\>0 ̯\>0 ^>0 0 4"","H"^>00^>00C _'8@0(  B S  ?!n2B8*urn:schemas-microsoft-com:office:smarttagsdate 1120048DayMonthYear'0KM  & ( ) , > A p s ^ ` 1':[]eg      n p   R T N O ce33333333333333333333;<xyz{|}&'01?@OWX[`1dhƅN OcV^`o() ^`hH.  L ^ `LhH. l l ^l `hH. <<^<`hH.  L ^ `LhH. ^`hH. ^`hH. |L|^|`LhH.^`o(.^`o() pLp^p`LhH. @ @ ^@ `hH. ^`hH. L^`LhH. ^`hH. ^`hH. PLP^P`LhH.^`o(. ^`hH. pLp^p`LhH. @ @ ^@ `hH. ^`hH. L^`LhH. ^`hH. ^`hH. PLP^P`LhH.NOcd ҈M         0.                  [=$?Or{oq`|+]7,7PO#u3@'' "''p@UnknownS: Times New RomanArial5Symbol3& : Arial"qh***  $  $!24d3QH)??O@ Statistics 101 Review Example on Bivariate March 17, 2004greenekrieger   Oh+'0 0 @L h t  A Statistics 101 Review Example on Bivariate March 17, 2004n greenetreereeNormaltkrieger2ieMicrosoft Word 10.0@@*K@RsL@RsL ՜.+,04 hp   Wharton-MKTGs$  A Statistics 101 Review Example on Bivariate March 17, 2004 Title  !"#$%&'()*+,-./12345679:;<=>?BRoot Entry F]LDData 1Table5 WordDocument",SummaryInformation(0DocumentSummaryInformation88CompObjj  FMicrosoft Word Document MSWordDocWord.Document.89q