INTRO TO PROBABILITY
INTRO TO PROBABILITY STA 4321
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Date Created: 09/18/15
Example Multinomial Distribution 2000 Election Setting In the 2000 election the percentage vote for BushGoreOther was 48484 Thus voters were classified as in the following based on probabilities proportions Candidate Probability Bush 48 Gore 48 Other 04 Suppose that n3000 voters were to be selected at random from this population and asked to reveal who they voted for 0 Give the distribution of the random variable X B X G X 0 O Give the expected values of XE XGXO I Give the variances and standard deviations of X B X g X 0 1 Give the covariances among all pairs of X B XGXO 00 Give the mean variance and standard deviation of X BX G gt Repeat the past 4 steps for the proportions where 17 Xn for each category candidate The shape of the distribution for the difference X BXG as well as the difference in their proportions is approximately normal Give a range of values that you would expect the difference in the sample proportions to lie in for approximately 95 of all samples of size n3000 Source National Council on Public Polls and virtually any election source BivorioTe Distributions SecTion 52 OfTen we ore inTeresTed in The ouTcomes of 2 or more rondom voriobles In The cose of Two rondom voriobles we will lobel Them X ond Y Suppose you hove The opporuniTy To purchase shores of Two firms Your subjecTive joinT probobiliTy disTribuTion pxy for The reTurn on The Two sTocllts is given below where pxy ProbXx ond Yy This is like on inTersecTion of evenTs in ChopTer 6 0SpxyS1 for alli ZZpxy1 way Stock B Return Y Stock A Return X 0 10 5 015 035 15 035 015 For insTonce The probobiliTy They boTh perform poorly X 5 ond YO is smoll Ol 5 Also The proboiliTy ThoT They boTh perform sTrongly Xl 5 ond Yi O is smoll Ol 5 lT s more likely ThoT one will perform sTrongly while The oTher will perform weolltly Xl 5 ond YO or X 5 ond Yi O eoch ouTcome wiTh probobiliTy 035 We con Think of These firms os subsTiTuTes Marginal DisTribuTions Ivlorginolly whoT is The probobiliTy disTribuTion for sTocllt A This is colled The morginol disTribuTion For sTocllt B These ore given in The following Toble ond ore compuTed by summing The joinT probobiliTies ocross The level of The oTher vorioble Siock A Siock B X plxlplx0lplx10 y plylpl5ylpli 5y 5 1535 50 0 1535 50 i5 35i5 50 10 35i5 50 Hence we con compuTe The meon ond vorionce forX ond Y X Expo 550 1550 50 a Xxx XV P 5 520515 5205 1000510005 100 y Zypoi 0501050 50 a ZO My My 2 0 520510 5205 2505 2505 25 So boTh sTocllts have The same expecTed reTLJrn bLJT sTocllt A is riskier in The sense ThaT iTs variance is much larger NoTe ThaT The sTandard deviaTions are The square rooTs of The variances ox l00 and or 50 How do X and Y quotco vary39 TogeTher Covariance C0VXY EX XXY y 2206 XyYPxy szypxy X y For These Two firms we find ThaT The covariance is negaTive since high values of X Tend To be seen wiTh low values of Y and vice verso We compUTe The Covariance of Their reTLJrns in The following Table 11x w 5 X y plXAl Xy X39MX y w Xy XAl lX39MXll y wlplxyl 5 0 15 0 lO 5 015o 1051575 5 TO 35 50 l 0 5 5035 l 75 l 0 5 35 l 75 15 0 35 0 TO 5 035o l 0 5 35 l 75 15 TO 15 150 TO 5 150l 225 1051575 Sum 50 200 COVXY 200 505050 The negaTive comes from The facT ThaT when X Tends To be large Y Tends To be small and vice versa based on The joinT probabiliTy disTribLJTion Coe icienT of CorrelaTion COVXY p lt 0X 0Y For The sTocllt daTa COVXY 200 ax 100 av 50 p 20105 2050 040 Functions of Random Variables Probability Distribution of the Sum of Two Variables Suppose you purchase i unit of each stock What is your expected return in percent You want the probability distribuion for the random variable XY Consider the joint probability distribution of X and Y and compute XY for each outcome X y 10 XIV xy xy10 XIV x210 XIV 5 0 15 50 5 515 075 25111 51375 5 10 35 5105 5135175 2535875 15 0 35 15015 1535525 225 357875 15 10 15 151025 2515375 625i59375 Sum 100 1000 18500 of ZZxy2pxy u y 18500 10002 8500 Thus the mean variance and standard deviation of XY the sum of the returns are i000 8500 and 922 respectively Rules for the Mean and Variance of XY EX Y N X 15 EltXgtEltYgt VX Y of VX VY 2COVX Y For the stock return example we have EX 5 EY 5 VX 100 VY 25 COVXY 20 Which gives us EXY 55 10 VXY lOO2522085 Which is in agreement With What we computed by generating the probability distribution for XY by brute force above Probability Distribution of a Linear Function of Two Variables Consider a portfolio of two stocks with Returns R1R2 and fixed weights W1W2 Return of portfolio Rp Wl Rt W2R2 Where WlW2 lWl20 W220 Expected Return on portfolio ERp Wi ERi W2ER2 Variance of Return on Portfolio VRp Wil2VRi W2l2VR2 2WiW2COVRiR2 Note that the rules for expected value and variance of linear functions does not depend on the weights summing to i For stock portfolio from two stocllts given above set R1 X and R2 Y Rp WiRi W2R2 WiRi l WilR2 Expected Return ERp W1 5 lWill5 5 Variance of Return VRp Wi2lOO lWi225 2WilWi2O Compute the variance if i Wl O25 and W2O75 ii Wl000 and W2l 00 iii Wll OO and W2000 To minimize the variance of returns we expand the equation above in terms of Wl take its derivative With respect to Wl set it equal to O and solve for WlI i VRpl65Wl2 90Wl 25 ii dVRpdwl 2l65Wl 90O iii 330Wl 90 gt WW 90330 02727 No mailer whoi porTfolio we choose expecied reTurns are 50 however we can minimize The variance of The reTurn risk by buying 027 pads of Siock A and 1 027073 pans of siock B A clossic poper on This Topic more moThemoTicolly rigorous Thon This exomple where eoch sTocllt hos only Two possible ouTcomes is given in Horry Ivl MorkowiTz PorTfolio SelecTionquot Journal of Finance 7 Morch l952 pp 77 9l