BHW #13 1/ Cooper. ENGR 323 Probabilistic Analysis Beautiful Homework # 13

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Darcy Knight
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1 BHW # /5 ENGR Probabilistic Aalysis Beautiful Homework # Three differet roads feed ito a particular freeway etrace. Suppose that durig a fixed time period, the umber of cars comig from each road oto the freeway is a radom variable, with the expected value ad stadard deviatio as give i the table. Table Give Expected Values ad Stadard Deviatios Road Road Road Expected value Stadard deviatio A. What is the expected total umber of cars eterig the freeway at this poit durig the period? (HINT: let X i = the umber from road i) B. What is the variace of the total umber of eterig cars? Have you made ay assumptios about the relatioship betwee the umbers of cars o the differet roads? C. With X i deotig the umber of cars eterig from road i durig the period, suppose that Cov (X, X ) = 80, Cov (X, X ) = 90, Cov (X, X ) = 00 (so that the three streams of traffic are ot idepedet). Compute the expected total umber of cars eterig cars ad the stadard deviatio of the total. Additioal Iformatio Before gettig started o the solutios I thik it is importat to defie a liear combiatio. I our text, o page 7, the defiitio for a liear combiatio is give as follows: Give a collectio of radom variables X, X,, X ad umerical costats a, a,, a the radom variable Y = a X + a X + + a X = Σ a i X i Is called a liear combiatio of the X i s Solutio A. The questio is askig for the total umber of cars eterig the freeway. The hit suggests that we let X i = the umber of cars from road i. This meas that X = the umber of cars from road, X = the umber of cars from road, ad X = the umber of cars from road. I Table the expected values for each road are give as follows: E(X ) = 800 E(X ) = 000 E(X ) = 600

2 BHW # /5 We kow from previous chapters that the expected value (E(X I )) is also called the mea value (µ I ). Aother way to look at the give values i the table is as follows: µ = 800 µ = 000 µ = 600 The propositio o page 8 of our text states: Let X, X,, X have mea values µ, µ,, µ respectively, ad variaces σ, σ,, σ respectively. Whether or ot the X I s are idepedet, E(a X + a X + + a X ) = a E(X ) + a E(X ) + + a E(X ) = a µ + a µ + + a µ Give the iformatioi the table, propositio ad the defiitio of a liear combiatio we ca ow solve part a. of this questio. For this liear combiatio we will assume that all of the a i s are equal to. To fid the total expected value of the liear combiatio all we eed to do is add the idividual expected values. a = a = a = µ = 800 µ = 000 µ = 600 E(a X + a X + a X ) = a E(X ) + a E(X ) + a E(X ) = a µ + a µ + a µ = ()(800) + ()(000) +()(600) = = 400 cars B. The questio is askig for the variace of the total umber of eterig cars. The questio is also askig what if ay assumptios are made i fidig the variace. As you might remember from previous chapters the variace [V(X i )] is the same as the stadard deviatio squared (σ I ). V(X i ) = σ I Table gives the followig values for the stadard deviatio. σ = 6 cars σ = 5 cars σ = 8 cars Sice the variace is the stadard deviatio squared (V(X i ) = σ I ) we ca calculate the variace. V(X ) = σ I V(X ) = σ V(X ) = σ V(X ) = (6 cars) V(X ) = (5 cars) V(X ) = (8 cars) V(X ) = 56 cars V(X ) = 65 cars V(X ) = 4 cars

3 BHW # /5 The propositio o page 8 of our text states: Let X, X,, X have mea values µ, µ,, µ respectively, ad variaces σ, σ,, σ respectively. If X, X,, X are idepedet, V(a X + a X + + a X ) = a V(X ) + a V(X ) +.+ a V(X ) = a σ + a σ + + a σ Now we are ready to solve part b of this questio. If we assume idepedece the we ca use the propositio o page 8 of our text. We will assume that all of the a i are equal to oe (a i = ). V(a X + a X + a X ) = a V(X ) + a V(X ) + a V(X ) = a σ + a σ + a σ = () (6) + () (5) + () (8) = = 05 cars C. This questio is askig for us to compute the expected total umber of eterig cars ad the stadard deviatio of the total. For this we are give ew iformatio. We are give values for the covariace. From page of our text we ca fid that the covariace is a measuremet of how strogly two radom variables are related to oe aother. The covariace betwee two radom variables X ad Y is: Cov(X, Y) = E[(X - µ x )( Y - µ y )] = ( x - µ x )( y - µ y )p(x,y) X ad Y are discrete X (x - µ x )( y - µ y )f(x,y)dx dy X ad Y are cotiuous The give values for the covariace for this questio are as follows: Cov(X, X ) = 80 cars Cov(X, X ) = 90 cars Cov(X, X ) = 00cars

4 BHW # 4/5 I our text o page 8 the followig propositio is give: Let X, X,, X have mea values µ, µ,, µ respectively, ad variaces σ, σ,, σ respectively. If X, X,, X are idepedet σ ax + ax + + ax = a σ + aσ a σ. For ay X, X,, X, V(a X + a X + + a X ) = i= j= a i a j Cov(X i, X j ) Now we have all of the ecessary iformatio to solve this questio. The expected value for the total umber of cars eterig will be the same: E(a X + a X + a X ) = a E(X ) + a E(X ) + a E(X ) = a µ + a µ + a µ = ()(800) + ()(000) +()(600) = = 400 cars The Variace for the total umber of cars will chage. We will still assume that all of the a i will be oe (a i = ). Usig part three of the propositio o page 8 of our book we ca solve for the variace usig the followig steps. Cov(X, X ) = 80 cars Cov(X, X ) = 90 cars Cov(X, X ) = 00cars V(X ) = 56 cars V(X ) = 65 cars V(X ) = 4 cars V(a X + a X + a X ) = i= j= a i a j Cov(X i, X j ) = Var(X ) + Var(X ) + Var(X ) + Cov(X + X ) + Cov(X + X ) + Cov(X + X ) = (80) + (90) + (00) = = 74 cars

5 BHW # 5/5 To fid the stadard deviatio of the total umber of cars we will use the secod part of the propositio o page 8 of our book. V(a X + a X + a X = a σ + a σ + a σ = 05 cars σ ax + ax + ax = a + σ + aσ aσ = 05cars = 4.7 cars

Background Information

Background Information Egieerig 33 Gayheart 5-63 1 Beautif ul Homework 5-63 Suppose t h a t whe t h e ph of a c e r t a i c h e m i c a l compoud i s 5.00, t h e ph measured by a radomly s e l e c t e d begiig chemistry studet