Home / Math and Physics Files / Math / Spectral Theory Book / Work for Aug 2013 Update / App G on random var

expectation value

PDF · 16 pages · 138.7 KB

Pages from a multivariate probability chapter (Chapter 5, sections 5.5-5.6, 'Attendance 9'), filed as an appendix on random variables in a spectral theory book update. It states the expectation and variance rules for functions of random variables, including linearity and independence. Multiple-choice exercises cover discrete fish waiting times, marbles drawn from an urn, and continuous potato-chip bag densities for two machines. The authorship is not shown in the text.

AI-written summary; may contain errors. This description is approximate.

Extracted text (machine-read; may contain errors)

Section5. TheExpectedValueofaFunctionofRandomVariables(AT TENDANCE9) 153 5.5 The Expected Value of a Function of Random Variables The material in this section is combined with the material in the next se ction and given in the next section. 5.6 Special Theorems ∙Expected value of a function of random variables, /u1D438[/u1D454(/u1D44C1,/u1D44C2,...,/u1D44C /u1D458)], equals =/uni007B.alt03/uni2211.alt01 all/u1D4661/uni2211.alt01 all/u1D4662⋅⋅⋅/uni2211.alt01 all/u1D466/u1D458/u1D454(/u1D4661,/u1D4662,...,/u1D466 /u1D458)/u1D45D(/u1D4661,/u1D4662,...,/u1D466 /u1D458) if discrete ,/uni222B.alt01∞ −∞/uni222B.alt01∞ −∞⋅⋅⋅/uni222B.alt01∞ −∞/u1D454(/u1D4661,/u1D4662,...,/u1D466 /u1D458)/u1D453(/u1D4661,/u1D4662,...,/u1D466 /u1D458)/u1D451/u1D4661/u1D451/u1D4662⋅⋅⋅/u1D451/u1D466/u1D458if continuous . ∙Also, the variance of a function of random variables, /u1D449[/u1D454(/u1D44C1,/u1D44C2,...,/u1D44C /u1D458)], with expected value, /u1D438[/u1D454(/u1D44C1,/u1D44C2,...,/u1D44C /u1D458)], is deﬁned /u1D449[/u1D454(/u1D44C1,/u1D44C2,...,/u1D44C /u1D458)] =/u1D438/uni005B.alt01 (/u1D454(/u1D44C1,/u1D44C2,...,/u1D44C /u1D458)−/u1D438[/u1D454(/u1D44C1,/u1D44C2,...,/u1D44C /u1D458)])2/uni005D.alt01 =/u1D438/uni005B.alt01 /u1D454(/u1D44C1,/u1D44C2,...,/u1D44C /u1D458)2/uni005D.alt01 −[/u1D438[/u1D454(/u1D44C1,/u1D44C2,...,/u1D44C /u1D458)]]2. ∙Related to this, /u1D438[/u1D44C1+/u1D44C2+⋅⋅⋅+/u1D44C/u1D458] =/u1D438[/u1D44C1]+⋅⋅⋅+/u1D438[/u1D44C/u1D458], /u1D438[/u1D4541(/u1D44C1,/u1D44C2)+⋅⋅⋅+/u1D454/u1D458(/u1D44C1,/u1D44C2)] =/u1D438[/u1D4541(/u1D44C1,/u1D44C2)]+⋅⋅⋅+/u1D438[/u1D454/u1D458(/u1D44C1,/u1D44C2)], /u1D438(/u1D450) =/u1D450,if/u1D450is a constant , /u1D438[/u1D450/u1D454(/u1D44C1,/u1D44C2)] =/u1D450/u1D438[/u1D454(/u1D44C1,/u1D44C2)],if/u1D450is a constant , /u1D438[/u1D454(/u1D44C1)ℎ(/u1D44C2)] =/u1D438[/u1D454(/u1D44C1)]/u1D438[ℎ(/u1D44C2)],if/u1D454(/u1D44C1),ℎ(/u1D44C2) independent . Exercise 5.6 (Special Theorems) 1.Discrete Expected Value Calculations: Waiting Times To Cat ch Fish. The joint density,/u1D45D(/u1D4661,/u1D4662), of the number of minutes waiting to catch the ﬁrstﬁsh,/u1D4661, andthe number of minutes waiting to catch the secondﬁsh,/u1D4662, is given below. /u1D4662↓/u1D4661→1 2 3 total 10.01 0.01 0.07 0.09 20.02 0.02 0.08 0.12 30.08 0.08 0.63 0.79 total 0.11 0.11 0.78 1.00 154 Chapter 5. Multivariate Probability Distributions (ATTENDANCE 9) (a) The expected average waiting time, /u1D454(/u1D4661,/u1D4662) =/u1D4661+/u1D4662 2, over two trips is /u1D438/uni005B.alt03/u1D44C1+/u1D44C2 2/uni005D.alt03 =3/uni2211.alt02 /u1D4661=13/uni2211.alt02 /u1D4662=1/uni0028.alt03/u1D4661+/u1D4662 2/uni0029.alt03 /u1D45D(/u1D4661,/u1D4662) =/uni0028.alt031+1 2/uni0029.alt03 (0.01)+/uni0028.alt031+2 2/uni0029.alt03 (0.02)+/uni0028.alt031+3 2/uni0029.alt03 (0.08) +/uni0028.alt032+1 2/uni0029.alt03 (0.01)+/uni0028.alt032+2 2/uni0029.alt03 (0.02)+/uni0028.alt032+3 2/uni0029.alt03 (0.08) +/uni0028.alt033+1 2/uni0029.alt03 (0.07)+/uni0028.alt033+2 2/uni0029.alt03 (0.08)+/uni0028.alt033+3 2/uni0029.alt03 (0.63) = (choose one) (i) 2.385(ii)2.685(iii)2.785. (Hint: Type 1 ,1,1,2,2,2,3,3,3 in/u1D43F1and 1,2,3,1,2,3,1,2,3 in/u1D43F2, deﬁne /u1D43F3=/u1D43F1+/u1D43F2 2, type 0.01,0.02,0.08,0.01,0.02,0.08,0.07,0.07,0.63 in/u1D43F4, deﬁne /u1D43F5=/u1D43F3×/u1D43F4, sum/u1D43F5using 1-Var Stats /u1D43F5, read/uni2211.alt01/u1D465= 2.685.) (b) The expected totalwaiting time, /u1D454(/u1D4661,/u1D4662) =/u1D4661+/u1D4662, over two trips is /u1D438[/u1D44C1+/u1D44C2] =3/uni2211.alt02 /u1D4661=13/uni2211.alt02 /u1D4662=1(/u1D4661+/u1D4662)/u1D45D(/u1D4661,/u1D4662) = (1+1)(0 .01)+(1+2)(0 .02)+(1+3)(0 .08) + (2+1)(0 .01)+(2+2)(0 .02)+(2+3)(0 .08) + (3+1)(0 .07)+(3+2)(0 .08)+(3+3)(0 .63) = (choose one) (i) 5.37(ii)6.37(iii)7.37. (Hint: Deﬁne /u1D43F3=/u1D43F1+/u1D43F2, deﬁne/u1D43F5=/u1D43F3×/u1D43F4, sum/u1D43F5using 1-Var Stats /u1D43F5, read/uni2211.alt01/u1D465= 5.37.) (c) Marginal probability function for /u1D44C1is given by column totals; in this case: /u1D46611 2 3 /u1D45D1(/u1D4661)0.11 0.11 0.78 so expected waiting time for ﬁrsttrip is /u1D438[/u1D44C1] =3/uni2211.alt02 /u1D4661=1/u1D4661/u1D45D1(/u1D4661) = (1)(0 .11)+(2)(0 .11)+(3)(0 .78) = (choose one) (i) 1.67(ii)2.67(iii)3.67. (d) Marginal probability function for /u1D44C2is given by row totals; in this case: /u1D46621 2 3 /u1D45D2(/u1D4662)0.09 0.12 0.79 Section 6. Special Theorems (ATTENDANCE 9) 155 so expected waiting time for secondtrip is /u1D438[/u1D44C2] =3/uni2211.alt02 /u1D4662=1/u1D4662/u1D45D2(/u1D4662) = (1)(0 .09)+(2)(0 .12)+(3)(0 .79) = (choose one) (i) 1.7(ii)2.7(iii)3.7. (e) Notice /u1D438[/u1D44C1+/u1D44C2] = 5.37 =/u1D438[/u1D44C1]+/u1D438[/u1D44C2] = 2.67+2.7 (choose one) (i) True (ii)False (f) Since /u1D438[/u1D44C2 2] =3/uni2211.alt02 /u1D4662=1/u1D4662 2/u1D45D2(/u1D4662) = (12)(0.09)+(22)(0.12)+(32)(0.79) = 7.68, thenvariance insecondwaiting time is /u1D449[/u1D44C2] =/u1D438/uni005B.alt01 /u1D44C2 2/uni005D.alt01 −[/u1D438[/u1D44C2]]2= 7.68−2.72= (choose one) (i) 0.29(ii)0.39(iii)0.49. 2.Discrete Expectation Value Calculations: Marbles In An Urn .Marbles chosen at random with outreplacement from an urn consisting of 8 blue and 6 black marbles. For /u1D456th marble chosen, /u1D44C/u1D456= 0 if blue and /u1D44C/u1D456= 1 if black. /u1D4662↓/u1D4661→blue, 0 black, 1 /u1D45D2(/u1D4662) blue, 08⋅7 14⋅136⋅8 14⋅138⋅7+6⋅8 14⋅13 black, 18⋅6 14⋅136⋅5 14⋅138⋅6+6⋅5 14⋅13 /u1D45D1(/u1D4661)8⋅7+8⋅6 14⋅136⋅8+6⋅5 14⋅131 (a) The expected value of /u1D454(/u1D4661,/u1D4662) =/u1D4661/u1D4662is /u1D438[/u1D44C1/u1D44C2] =1/uni2211.alt02 /u1D4661=01/uni2211.alt02 /u1D4662=0(/u1D4661/u1D4662)/u1D45D(/u1D4661,/u1D4662) = (0×0)/uni0028.alt038⋅7 14⋅13/uni0029.alt03 +(0×1)/uni0028.alt038⋅6 14⋅13/uni0029.alt03 +(1×0)/uni0028.alt038⋅8 14⋅13/uni0029.alt03 +(1×1)/uni0028.alt036⋅5 14⋅13/uni0029.alt03 = (choose one) (i)8⋅6 14⋅13(ii)8⋅7 14⋅13(iii)6⋅5 14⋅13. 156 Chapter 5. Multivariate Probability Distributions (ATTENDANCE 9) (b) Expected value of /u1D44C1is /u1D438[/u1D44C1] =1/uni2211.alt02 /u1D4661=0/u1D4661/u1D45D1(/u1D4661) = (0)/uni0028.alt038⋅7+8⋅6 14⋅13/uni0029.alt03 +(1)/uni0028.alt036⋅8+6⋅5 14⋅13/uni0029.alt03 = (choose one) (i)8⋅7+8⋅6 14⋅13(ii)8⋅7+2×8⋅6 14⋅13(iii)6⋅8+6⋅5 14⋅13. (c) Expected value of /u1D44C2is /u1D438[/u1D44C2] =1/uni2211.alt02 /u1D4662=0/u1D4662/u1D45D2(/u1D4662) = (0)/uni0028.alt038⋅7+6⋅8 14⋅13/uni0029.alt03 +(1)/uni0028.alt038⋅6+6⋅5 14⋅13/uni0029.alt03 = (choose one) (i)8⋅7+8⋅6 14⋅13(ii)8⋅7+2×8⋅6 14⋅13(iii)8⋅6+6⋅5 14⋅13. (Notice, in this case, /u1D438[/u1D44C1] =/u1D438[/u1D44C2].) (d) In this case, /u1D438[/u1D44C1/u1D44C2] =6⋅5 14⋅13∕=/u1D438[/u1D44C1]/u1D438[/u1D44C2] =/uni0028.alt036⋅8+6⋅5 14⋅13/uni0029.alt03/uni0028.alt038⋅6+6⋅5 14⋅13/uni0029.alt03 because/u1D44C1and/u1D44C2aredependent . (choose one) (i) True (ii)False (e) Using the properties of expectation above, /u1D438[3/u1D44C1−2/u1D44C2] = 3/u1D438[/u1D44C1]−2/u1D438[/u1D44C2] = 3/uni0028.alt036⋅8+6⋅5 14⋅13/uni0029.alt03 −2/uni0028.alt038⋅6+6⋅5 14⋅13/uni0029.alt03 = (i)8⋅7+8⋅6 14⋅13(ii)8⋅7+2×8⋅6 14⋅13(iii)8⋅6+6⋅5 14⋅13. (f) Since /u1D438[/u1D44C2 2] =0/uni2211.alt02 /u1D4662=0/u1D4662 2/u1D45D2(/u1D4662) = (02)/uni0028.alt038⋅7+6⋅8 14⋅13/uni0029.alt03 +(12)/uni0028.alt038⋅6+6⋅5 14⋅13/uni0029.alt03 =8⋅6+6⋅5 14⋅13, then the variance is /u1D449[/u1D44C2] =/u1D438/uni005B.alt01 /u1D44C2 2/uni005D.alt01 −[/u1D438[/u1D44C2]]2=8⋅6+6⋅5 14⋅13−/uni0028.alt038⋅6+6⋅5 14⋅13/uni0029.alt032 ≈ (choose one) (i) 0.245(ii)0.345(iii)0.445. (g) The deﬁnition of variance is /u1D449[/u1D44C2] =/u1D438/uni005B.alt01 (/u1D44C2−/u1D438[/u1D44C2])])2/uni005D.alt01 =1/uni2211.alt02 /u1D4662=0(/u1D44C2−/u1D438[/u1D44C2])2/u1D45D2(/u1D4662) =/uni0028.alt03 0−8⋅6+6⋅5 14⋅13/uni0029.alt032/uni0028.alt038⋅7+6⋅8 14⋅13/uni0029.alt03 +/uni0028.alt03 1−8⋅6+6⋅5 14⋅13/uni0029.alt032/uni0028.alt038⋅6+6⋅5 14⋅13/uni0029.alt03 ≈ (choose one) (i) 0.245(ii)0.345(iii)0.445. Section 6. Special Theorems (ATTENDANCE 9) 157 3.Continuous Expectation Value Calculations: Potato Chips. Although each bag should weigh 50 grams each and contain 5 milligrams of salt, in fact, bec ause of diﬀeringmachines, weightandamountofsaltplacedineachbagvaries according to two probability functions below. (a)Machine A. Bivariate density function for machine A is /u1D453(/u1D4661,/u1D4662) =/uni007B.alt031 12,49≤/u1D4661≤51,2≤/u1D4662≤8 0 elsewhere i. The expected value of /u1D454(/u1D4661,/u1D4662) =/u1D4661/u1D4662is /u1D438[/u1D44C1/u1D44C2] =/uni222B.alt028 2/uni222B.alt0251 49(/u1D4661/u1D4662)/u1D453(/u1D4661,/u1D4662)/u1D451/u1D4661/u1D451/u1D4662 =/uni222B.alt028 2/uni222B.alt0251 49(/u1D4661/u1D4662)1 12/u1D451/u1D4661/u1D451/u1D4662 =/uni222B.alt028 21 12/u1D4662/uni0028.alt031 2/u1D4662 1/uni0029.alt03/u1D4661=51 /u1D4661=49/u1D451/u1D4662 =/uni0028.alt03100 24/u1D4662 2/uni0029.alt03/u1D4661=8 /u1D4661=2= (choose one) (i) 200(ii)250(iii)300. ii. Since marginal, /u1D4531(/u1D4661) =/uni222B.alt02∞ −∞/u1D453(/u1D4661,/u1D4662)/u1D451/u1D4662=/uni222B.alt028 21 12/u1D451/u1D4662=1 12(/u1D4662)/u1D4662=8 /u1D4662=2=1 2, 49≤/u1D4661≤51, expected value of /u1D44C1is /u1D438[/u1D44C1] =/uni222B.alt0251 49/u1D4661/u1D4531(/u1D4661)/u1D451/u1D4661=/uni222B.alt0251 49/u1D46611 2/u1D451/u1D4661=/uni0028.alt031 4/u1D4662 1/uni0029.alt03/u1D4662=51 /u1D4662=49= (choose one) (i) 50(ii)100(iii)150. iii. Since marginal, /u1D4532(/u1D4662) =/uni222B.alt02∞ −∞/u1D453(/u1D4661,/u1D4662)/u1D451/u1D4661=/uni222B.alt0251 491 12/u1D451/u1D4661=1 12(/u1D4661)/u1D4661=51 /u1D4661=49=1 6, 2≤/u1D4662≤8, expected value of /u1D44C2is /u1D438[/u1D44C2] =/uni222B.alt028 2/u1D4662/u1D4532(/u1D4662)/u1D451/u1D4662=/uni222B.alt028 2/u1D46621 6/u1D451/u1D4662=/uni0028.alt031 12/u1D4662 2/uni0029.alt03/u1D4662=8 /u1D4661=2= (choose one) (i) 5(ii)10(iii)15. 158 Chapter 5. Multivariate Probability Distributions (ATTENDANCE 9) iv. Since /u1D438[/u1D44C1/u1D44C2] = 250 = /u1D438[/u1D44C1]/u1D438[/u1D44C2] = (50)(5) /u1D44C1and/u1D44C2are (choose one) (i) dependent (ii)independent . v. Using the properties of expectation above, /u1D438[3/u1D44C1−2/u1D44C2] = 3/u1D438[/u1D44C1]−2/u1D438[/u1D44C2] = 3(50) −2(5) = (i)130(ii)140(iii)150. vi. Since /u1D438[/u1D44C2 2] =/uni222B.alt028 2/u1D4662 2/u1D4532(/u1D4662)/u1D451/u1D4662=/uni222B.alt028 2/u1D4662 21 6/u1D451/u1D4662=/uni0028.alt031 18/u1D4663 2/uni0029.alt03/u1D4662=8 /u1D4662=2= 28, then variance is /u1D449[/u1D44C2] =/u1D438/uni005B.alt01 /u1D44C2 2/uni005D.alt01 −[/u1D438[/u1D44C2]]2= 28−(5)2≈ (choose one) (i) 1(ii)2(iii)3. (b)Machine B. Bivariate density function for machine B is /u1D453(/u1D4661,/u1D4662) =/uni007B.alt031 4,49≤/u1D4661≤51,4≤/u1D4662≤6 0 elsewhere i. The expected value of /u1D454(/u1D4661,/u1D4662) =/u1D4661/u1D4662is /u1D438[/u1D44C1/u1D44C2] =/uni222B.alt026 4/uni222B.alt0251 49(/u1D4661/u1D4662)/u1D453(/u1D4661,/u1D4662)/u1D451/u1D4661/u1D451/u1D4662 =/uni222B.alt026 4/uni222B.alt0251 49(/u1D4661/u1D4662)1 4/u1D451/u1D4661/u1D451/u1D4662 =/uni222B.alt026 41 4/u1D4662/uni0028.alt031 2/u1D4662 1/uni0029.alt03/u1D4661=51 /u1D4661=49/u1D451/u1D4662 =/uni0028.alt03100 8/u1D4662 2/uni0029.alt03/u1D4661=6 /u1D4661=4= (choose one) (i) 250(ii)500(iii)750. ii. Since marginal, /u1D4531(/u1D4661) =/uni222B.alt02∞ −∞/u1D453(/u1D4661,/u1D4662)/u1D451/u1D4662=/uni222B.alt026 41 4/u1D451/u1D4662=1 4(/u1D4662)/u1D4662=6 /u1D4662=4=1 2, 49≤/u1D4661≤51, expected value of /u1D44C1is /u1D438[/u1D44C1] =/uni222B.alt0251 49/u1D4661/u1D4531(/u1D4661)/u1D451/u1D4661=/uni222B.alt0251 49/u1D46611 2/u1D451/u1D4661=/uni0028.alt031 4/u1D4662 1/uni0029.alt03/u1D4662=51 /u1D4662=49= (choose one) (i) 50(ii)100(iii)150. Section 6. Special Theorems (ATTENDANCE 9) 159 iii. Since marginal, /u1D4532(/u1D4662) =/uni222B.alt02∞ −∞/u1D453(/u1D4661,/u1D4662)/u1D451/u1D4661=/uni222B.alt0251 491 4/u1D451/u1D4661=1 4(/u1D4661)/u1D4661=51 /u1D4661=49=1 2, 4≤/u1D4662≤6, expected value of /u1D44C2is /u1D438[/u1D44C2] =/uni222B.alt026 4/u1D4662/u1D4532(/u1D4662)/u1D451/u1D4662=/uni222B.alt026 4/u1D46621 2/u1D451/u1D4662=/uni0028.alt031 4/u1D4662 2/uni0029.alt03/u1D4662=6 /u1D4662=4= (choose one) (i) 5(ii)10(iii)15. iv. Since /u1D438[/u1D44C1/u1D44C2] = 250 = /u1D438[/u1D44C1]/u1D438[/u1D44C2] = (50)(5) /u1D44C1and/u1D44C2are (choose one) (i) dependent (ii)independent . v. Using the properties of expectation above, /u1D438[3/u1D44C1+3/u1D44C2] = 3/u1D438[/u1D44C1]+3/u1D438[/u1D44C2] = 3(50)+3(5) = (i)155(ii)160(iii)165. vi. Since /u1D438[/u1D44C2 2] =/uni222B.alt026 4/u1D4662 2/u1D4532(/u1D4662)/u1D451/u1D4662=/uni222B.alt026 4/u1D4662 21 2/u1D451/u1D4662=/uni0028.alt031 6/u1D4663 2/uni0029.alt03/u1D4662=6 /u1D4662=4=76 3, then variance is /u1D449[/u1D44C2] =/u1D438/uni005B.alt01 /u1D44C2 2/uni005D.alt01 −[/u1D438[/u1D44C2]]2=76 3−(5)2≈ (choose one) (i)1 3(ii)2 3(iii)3 3. vii. Let/u1D448=/u1D44C1−/u1D44C2and assume /u1D44C1and/u1D44C2are independent. /u1D449(/u1D448) =/u1D438(/u1D4482)−/u1D438(/u1D448)2 =/u1D438/uni005B.alt01 (/u1D44C1−/u1D44C2)2/uni005D.alt01 −[/u1D438(/u1D44C1−/u1D44C2)]2 =/u1D438/uni005B.alt01 /u1D44C2 1−2/u1D44C1/u1D44C2+/u1D44C2 2/uni005D.alt01 −[/u1D438(/u1D44C1)−/u1D438(/u1D44C2)]2 =/u1D438/uni005B.alt01 /u1D44C2 1/uni005D.alt01 −2/u1D438[/u1D44C1/u1D44C2]+/u1D438/uni005B.alt01 /u1D44C2 2/uni005D.alt01 −/uni007B.alt01 [/u1D438(/u1D44C1)]2−2/u1D438[/u1D44C1]/u1D438[/u1D44C2]+[/u1D438(/u1D44C2)]2/uni007D.alt01 =/u1D438/uni005B.alt01 /u1D44C2 1/uni005D.alt01 −2/u1D438[/u1D44C1/u1D44C2]+/u1D438/uni005B.alt01 /u1D44C2 2/uni005D.alt01 −/uni007B.alt01 [/u1D438(/u1D44C1)]2−2/u1D438[/u1D44C1/u1D44C2]+[/u1D438(/u1D44C2)]2/uni007D.alt01 =/u1D438/uni005B.alt01 /u1D44C2 1/uni005D.alt01 −[/u1D438(/u1D44C1)]2+/uni007B.alt01 /u1D438/uni005B.alt01 /u1D44C2 2/uni005D.alt01 −[/u1D438(/u1D44C2)]2/uni007D.alt01 =/u1D449(/u1D44C1)+/u1D449(/u1D44C2) (i)True (ii)False 160 Chapter 5. Multivariate Probability Distributions (ATTENDANCE 9) 4.More Continuous Expectation Value Calculations. Consider /u1D453(/u1D4661,/u1D4662) =/uni007B.alt036 5(2−/u1D4661)(1−/u1D4662) 0≤/u1D4661≤2,0≤/u1D4662≤1, /u1D4661+2/u1D4662<2 0 elsewhere (a) Since /u1D4661+2/u1D4662≤2, then/u1D4662≤1−1 2/u1D4661, and marginal of /u1D44C1is 6 5/uni222B.alt021−1 2/u1D4661 0(2−/u1D4661)(1−/u1D4662)/u1D451/u1D4662=6 5/uni222B.alt021−1 2/u1D4661 0(2−/u1D4661−2/u1D4662+/u1D4661/u1D4662)/u1D451/u1D4662 =6 5/uni0028.alt03 2/u1D4662−/u1D4661/u1D4662−/u1D4662 2+1 2/u1D4661/u1D4662 2/uni0029.alt03/u1D4662=1−1 2/u1D4661 /u1D4662=0 =6 5/uni0028.alt03 1−1 2/u1D4661−1 4/u1D4662 1+1 8/u1D4663 1/uni0029.alt03 where 0≤/u1D4661≤2. So, the expected value of /u1D44C1is /u1D438[/u1D44C1] =/uni222B.alt022 0/u1D4661/u1D4531(/u1D4661)/u1D451/u1D4661=6 5/uni222B.alt022 0/u1D4661/uni0028.alt03 1−1 2/u1D4661−1 4/u1D4662 1+1 8/u1D4663 1/uni0029.alt03 /u1D451/u1D4661 =6 5/uni0028.alt031 2/u1D4662 1−1 6/u1D4663 1−1 16/u1D4664 1+1 40/u1D4665 1/uni0029.alt03/u1D4662=2 /u1D4662=0= (choose one) (i)13 25(ii)14 25(iii)15 25. (b) Since /u1D4661+2/u1D4662≤2, then/u1D4661≤2−2/u1D4662, and marginal of /u1D44C2is 6 5/uni222B.alt022−2/u1D4662 0(2−/u1D4661)(1−/u1D4662)/u1D451/u1D4661=6 5/uni222B.alt022−2/u1D4662 0(2−/u1D4661−2/u1D4662+/u1D4661/u1D4662)/u1D451/u1D4661 =6 5/uni0028.alt03 2/u1D4661−1 2/u1D4662 1−2/u1D4661/u1D4662+1 2/u1D4662 1/u1D4662/uni0029.alt03/u1D4661=2−2/u1D4662 /u1D4661=0 =6 5/uni0028.alt01 2−2/u1D4662−2/u1D4662 2+2/u1D4663 2/uni0029.alt01 where 0≤/u1D4661≤1. So, the expected value of /u1D44C2is /u1D438[/u1D44C2] =/uni222B.alt021 0/u1D4662/u1D4532(/u1D4662)/u1D451/u1D4662=6 5/uni222B.alt021 0/u1D4662/uni0028.alt01 2−2/u1D4662−2/u1D4662 2+2/u1D4663 2/uni0029.alt01 /u1D451/u1D4662 =6 5/uni0028.alt03 /u1D4662 2−2 3/u1D4663 2−2 4/u1D4664 2+2 5/u1D4665 2/uni0029.alt03/u1D4662=1 /u1D4662=0= (choose one) (i)7 25(ii)8 25(iii)9 25. Section 6. Special Theorems (ATTENDANCE 9) 161 (c) Using the properties of expectation above, /u1D438[25/u1D44C1+50/u1D44C2] = 25/u1D438[/u1D44C1]+50/u1D438[/u1D44C2] = 25/uni0028.alt0317 25/uni0029.alt03 +50/uni0028.alt037 25/uni0029.alt03 = (i)29(ii)30(iii)31. (d) Since /u1D438[/u1D44C2 2] =/uni222B.alt021 0/u1D4662 2/u1D4532(/u1D4662)/u1D451/u1D4662=6 5/uni222B.alt021 0/u1D4662 2/uni0028.alt01 2−2/u1D4662−2/u1D4662 2+2/u1D4663 2/uni0029.alt01 /u1D451/u1D4662 =6 5/uni0028.alt032 3/u1D4663 2−2 4/u1D4664−2 5/u1D4665 2+2 6/u1D4666 2/uni0029.alt03/u1D4662=1 /u1D4662=0=3 25, then variance is /u1D449[/u1D44C2] =/u1D438/uni005B.alt01 /u1D44C2 2/uni005D.alt01 −[/u1D438[/u1D44C2]]2=3 25−/uni0028.alt037 25/uni0029.alt032 ≈ (choose one) (i)24 625(ii)25 625(iii)26 625. (e) According to Tchebysheﬀ’s theorem, the probability variable /u1D44C2iswithin /u1D458= 3 standard deviations of mean /u1D438[/u1D44C2] isat least /u1D443(∣/u1D44C−/u1D707∣< /u1D458/u1D70E)≥1−1 /u1D4582= 1−1 32=8 9. That is, there is at least a8 9th chance the following interval contains /u1D44C2: /u1D438[/u1D44C2]±3/uni221A.alt01 /u1D449[/u1D44C2] =7 25±3/uni221A.alt03 26 625≈ (i)(−0.23,0.79)(ii)(−0.33,0.89)(iii)(−0.43,0.99). 5.Expected Number of Matches. Ten people throw ten tickets with their names on each ticket into a jar, then draw one ticket out of the jar at rando m (and put it back in the jar). Let /u1D44Bbe the number of people who select their own ticket out of the jar. Let /u1D44B=/u1D44C1+/u1D44C2+⋅⋅⋅+/u1D44C10 where /u1D44C/u1D456=/uni007B.alt03 1 if/u1D456th person selects own ticket 0 if/u1D456th person does not select their own ticket (a) Each person chooses any of the ten tickets with equal chance , /u1D438[/u1D44C/u1D456] = 1×/u1D45D(1)+0×/u1D45D(0) = (choose one) (i)1 10(ii)2 10(iii)3 10. (b) Expected number of ten individuals to choose their own ticket is /u1D438(/u1D44B) =/u1D438(/u1D44C1)+⋅⋅⋅+/u1D438(/u1D44C10) = 10×1 10= (i)8 10(ii)9 10(iii)10 10. We would expect one of ten individuals to choose their own ticket. (c)If/u1D45Bindividuals played this game, then we would expect /u1D438(/u1D44B) =/u1D438(/u1D44C1)+⋅⋅⋅+/u1D438(/u1D44C/u1D45B) =/u1D45B/uni0028.alt011 /u1D45B/uni0029.alt01 = (i)/u1D48F−1 /u1D48F(ii)/u1D48F /u1D48F(iii)/u1D48F+1 /u1D48F. 162 Chapter 5. Multivariate Probability Distributions (ATTENDANCE 9) 5.7 The Covariance of Two Random Variables We look at covariance , a measure of how two random variables are linearly related. Covariance is deﬁned by Cov(/u1D44C1,/u1D44C2) =/u1D438[(/u1D44C1−/u1D438(/u1D44C2))(/u1D44C2−/u1D438(/u1D44C2))] =/u1D438(/u1D44C1/u1D44C2)−/u1D438(/u1D44C1)/u1D438(/u1D44C2), and has the following properties, ∙Cov(/u1D44C1,/u1D44C2) = Cov( /u1D44C2,/u1D44C1), ∙Cov(/u1D44C1,/u1D44C1) =/u1D449(/u1D44C1), ∙Cov(/u1D44E/u1D44C1,/u1D44C2) =/u1D44ECov(/u1D44C1,/u1D44C2), where /u1D44Eis a constant, ∙Cov(/u1D44C1,/u1D44C2) = 0 if/u1D44C1,/u1D44C2are uncorrelated or independent. Uncorrelated random variables are notnecessarily independent random variables. Independent random variables arenecessarily uncorrelated random variables. The correlation /u1D70C,−1≤/u1D70C≤1, is given by /u1D70C(/u1D4661,/u1D4662) =/u1D70C=Cov(/u1D44C1,/u1D44C2)/uni221A.alt01 /u1D449(/u1D44C1)/u1D449(/u1D44C2)=Cov(/u1D44C1,/u1D44C2) /u1D70E1/u1D70E2. Exercise 5.7 (The Covariance of Two Random Variables) 1.Covariance and Correlation: Waiting Times To Catch Fish. The joint density, /u1D45D(/u1D4661,/u1D4662), of the number of minutes waiting to catch the ﬁrstﬁsh,/u1D4661,andthe number of minutes waiting to catch the secondﬁsh,/u1D4662, is given below. /u1D4662↓/u1D4661→1 2 3 total 10.01 0.01 0.07 0.09 20.02 0.02 0.08 0.12 30.08 0.08 0.63 0.79 total 0.11 0.11 0.78 1.00 (a) If/u1D454(/u1D4661,/u1D4662) =/u1D4661/u1D4662, then /u1D438[/u1D44C1/u1D44C2] =3/uni2211.alt02 /u1D4661=13/uni2211.alt02 /u1D4662=1(/u1D4661/u1D4662)/u1D45D(/u1D4661,/u1D4662) = (1×1)(0.01)+(1×2)(0.02)+(1×3)(0.08) + (2×1)(0.01)+(2×2)(0.02)+(2×3)(0.08) + (3×1)(0.07)+(3×2)(0.08)+(3×3)(0.63) = Section 7. The Covariance of Two Random Variables (ATTENDANCE 9) 163 (choose one) (i) 5.23(ii)6.23(iii)7.23. (Hint: Type 1 ,1,1,2,2,2,3,3,3 in/u1D43F1and 1,2,3,1,2,3,1,2,3 in/u1D43F2, deﬁne /u1D43F3=/u1D43F1×/u1D43F2, type 0.01,0.02,0.08,0.01,0.02,0.08,0.07,0.08,0.63 in/u1D43F4, deﬁne /u1D43F5=/u1D43F3×/u1D43F4, sum/u1D43F5using 1-Var Stats /u1D43F5, read/uni2211.alt01/u1D465= 7.23.) (b) Since /u1D438[/u1D44C1] =3/uni2211.alt02 /u1D4661=1/u1D4661/u1D45D1(/u1D4661) = (1)(0 .11)+(2)(0 .11)+(3)(0 .78) = 2.67 /u1D438[/u1D44C2] =3/uni2211.alt02 /u1D4662=1/u1D4662/u1D45D2(/u1D4662) = (1)(0 .09)+(2)(0 .12)+(3)(0 .79) = 2.7, then covariance is Cov(/u1D44C1,/u1D44C2) =/u1D438(/u1D44C1/u1D44C2)−/u1D438(/u1D44C1)/u1D438(/u1D44C2) = 7.23−(2.67)(2.7)≈ (choose one) (i) −0.039(ii)0.021(iii)0.139. (c) Since /u1D438[/u1D44C2 1] =3/uni2211.alt02 /u1D4661=1/u1D4662 1/u1D45D1(/u1D4661) = (12)(0.11)+(22)(0.11)+(32)(0.78) = 7.57, /u1D449[/u1D44C1] =/u1D438/uni005B.alt01 /u1D44C2 1/uni005D.alt01 −[/u1D438[/u1D44C1]]2= 7.57−2.672= 0.4411, /u1D438[/u1D44C2 2] =3/uni2211.alt02 /u1D4662=1/u1D4662 2/u1D45D2(/u1D4662) = (12)(0.09)+(22)(0.12)+(32)(0.79) = 2.7, /u1D449[/u1D44C2] =/u1D438/uni005B.alt01 /u1D44C2 2/uni005D.alt01 −[/u1D438[/u1D44C2]]2= 7.68−2.72= 0.39, then correlation is /u1D70C=Cov(/u1D44C1,/u1D44C2)/uni221A.alt01 /u1D449(/u1D44C1)/u1D449(/u1D44C2)=−0.039√0.4411×0.39≈ (choose one) (i) −0.235(ii)−0.139(iii)−0.094. There is very little linear correlation between two waiting times. (d) Let/u1D4481=/u1D44C1+/u1D44C2and/u1D4482=/u1D44C1−/u1D44C2. Then Cov(/u1D4481,/u1D4482) =/u1D438(/u1D4481/u1D4482)−/u1D438(/u1D4481)/u1D438(/u1D4482) =/u1D438[(/u1D44C1+/u1D44C2)(/u1D44C1−/u1D44C2)]−/u1D438((/u1D44C1+/u1D44C2))/u1D438((/u1D44C1−/u1D44C2)) =/u1D438/uni005B.alt01 /u1D44C2 1−/u1D44C2 2/uni005D.alt01 −[/u1D438(/u1D44C1)+/u1D438(/u1D44C2)][/u1D438(/u1D44C1)−/u1D438(/u1D44C2)] =/u1D438/uni005B.alt01 /u1D44C2 1/uni005D.alt01 −/u1D438/uni005B.alt01 /u1D44C2 2/uni005D.alt01 −/uni007B.alt01 [/u1D438(/u1D44C1)]2−[/u1D438(/u1D44C2)]2/uni007D.alt01 =/u1D438/uni005B.alt01 /u1D44C2 1/uni005D.alt01 −[/u1D438(/u1D44C1)]2−/uni007B.alt01 /u1D438/uni005B.alt01 /u1D44C2 2/uni005D.alt01 −[/u1D438(/u1D44C2)]2/uni007D.alt01 =/u1D449(/u1D44C1)−/u1D449(/u1D44C2) = 0.4411−0.39≈ (choose one) (i) 0.0355(ii)0.0392(iii)0.0511. 164 Chapter 5. Multivariate Probability Distributions (ATTENDANCE 9) 2.Covariance and Correlation: Marbles In An Urn. Marbles chosen at random withoutreplacement from an urn consisting of 8 blue and 6 black marbles. For /u1D456th marble chosen, /u1D44C/u1D456= 0 if blue and /u1D44C/u1D456= 1 if black. /u1D4662↓/u1D4661→blue, 0 black, 1 /u1D45D2(/u1D4662) blue, 08⋅7 14⋅136⋅8 14⋅138⋅7+6⋅8 14⋅13 black, 18⋅6 14⋅136⋅5 14⋅138⋅6+6⋅5 14⋅13 /u1D45D1(/u1D4661)8⋅7+8⋅6 14⋅136⋅8+6⋅5 14⋅131 (a) Since /u1D438[/u1D44C1/u1D44C2] =1/uni2211.alt02 /u1D4661=01/uni2211.alt02 /u1D4662=0(/u1D4661/u1D4662)/u1D45D(/u1D4661,/u1D4662) = (0×0)/uni0028.alt038⋅7 14⋅13/uni0029.alt03 +(0×1)/uni0028.alt038⋅6 14⋅13/uni0029.alt03 +(1×0)/uni0028.alt038⋅8 14⋅13/uni0029.alt03 +(1×1)/uni0028.alt036⋅5 14⋅13/uni0029.alt03 =6⋅5 14⋅13, /u1D438[/u1D44C1] =1/uni2211.alt02 /u1D4661=0/u1D4661/u1D45D1(/u1D4661) = (0)/uni0028.alt038⋅7+8⋅6 14⋅13/uni0029.alt03 +(1)/uni0028.alt036⋅8+6⋅5 14⋅13/uni0029.alt03 =6⋅8+6⋅5 14⋅13, /u1D438[/u1D44C2] =1/uni2211.alt02 /u1D4662=0/u1D4662/u1D45D2(/u1D4662) = (0)/uni0028.alt038⋅7+6⋅8 14⋅13/uni0029.alt03 +(1)/uni0028.alt038⋅6+6⋅5 14⋅13/uni0029.alt03 =8⋅6+6⋅5 14⋅13, then covariance is Cov(/u1D44C1,/u1D44C2) =/u1D438(/u1D44C1/u1D44C2)−/u1D438(/u1D44C1)/u1D438(/u1D44C2) = (choose one) (i) −12 637(ii)−13 637(iii)−14 637. Section 7. The Covariance of Two Random Variables (ATTENDANCE 9) 165 (b) Since /u1D438[/u1D44C2 1] =3/uni2211.alt02 /u1D4661=1/u1D4662 1/u1D45D1(/u1D4661) = (02)/uni0028.alt038⋅7+8⋅6 14⋅13/uni0029.alt03 +(12)/uni0028.alt036⋅8+6⋅5 14⋅13/uni0029.alt03 =6⋅8+6⋅5 14⋅13, /u1D449[/u1D44C1] =/u1D438/uni005B.alt01 /u1D44C2 1/uni005D.alt01 −[/u1D438[/u1D44C1]]2=6⋅8+6⋅5 14⋅13−/uni0028.alt038⋅6+6⋅5 14⋅13/uni0029.alt032 ≈0.245, /u1D438[/u1D44C2 2] =0/uni2211.alt02 /u1D4662=0/u1D4662 2/u1D45D2(/u1D4662) = (02)/uni0028.alt038⋅7+6⋅8 14⋅13/uni0029.alt03 +(12)/uni0028.alt038⋅6+6⋅5 14⋅13/uni0029.alt03 =8⋅6+6⋅5 14⋅13, /u1D449[/u1D44C2] =/u1D438/uni005B.alt01 /u1D44C2 2/uni005D.alt01 −[/u1D438[/u1D44C2]]2=8⋅6+6⋅5 14⋅13−/uni0028.alt038⋅6+6⋅5 14⋅13/uni0029.alt032 ≈0.245, then correlation is /u1D70C=Cov(/u1D44C1,/u1D44C2)/uni221A.alt01 /u1D449(/u1D44C1)/u1D449(/u1D44C2)=−12 637√0.245×0.245≈ (choose one) (i) −0.077(ii)−0.139(iii)−0.294. (c)Relationship between covariance and variance. Cov(/u1D44C1,/u1D44C1) =/u1D438(/u1D44C1/u1D44C1)−/u1D438(/u1D44C1)/u1D438(/u1D44C1) =/u1D438(/u1D44C2 1)−[/u1D438(/u1D44C1)]2=/u1D449(/u1D4491)≈0.245 Cov(/u1D44C2,/u1D44C2) =/u1D438(/u1D44C2/u1D44C2)−/u1D438(/u1D44C2)/u1D438(/u1D44C2) =/u1D438(/u1D44C2 2)−[/u1D438(/u1D44C2)]2=/u1D449(/u1D4492)≈0.245 (i)True (ii)False 3.Covariance and Correlation: Weight and Amount of Salt in Pot ato Chips. (a)Machine A. Bivariate density function for machine A is /u1D453(/u1D4661,/u1D4662) =/uni007B.alt031 12,49≤/u1D4661≤51,2≤/u1D4662≤8 0 elsewhere 166 Chapter 5. Multivariate Probability Distributions (ATTENDANCE 9) i. Since /u1D438[/u1D44C1/u1D44C2] =/uni222B.alt028 2/uni222B.alt0251 49(/u1D4661/u1D4662)/u1D453(/u1D4661,/u1D4662)/u1D451/u1D4661/u1D451/u1D4662 =/uni222B.alt028 2/uni222B.alt0251 49(/u1D4661/u1D4662)1 12/u1D451/u1D4661/u1D451/u1D4662=/uni222B.alt028 21 12/u1D4662/uni0028.alt031 2/u1D4662 1/uni0029.alt03/u1D4661=51 /u1D4661=49/u1D451/u1D4662 =/uni0028.alt03100 24/u1D4662 2/uni0029.alt03/u1D4662=8 /u1D4662=2= 250, /u1D4531(/u1D4661) =/uni222B.alt02∞ −∞/u1D453(/u1D4661,/u1D4662)/u1D451/u1D4662 =/uni222B.alt028 21 12/u1D451/u1D4662=1 12(/u1D4662)/u1D4662=8 /u1D4662=2=1 2, /u1D438[/u1D44C1] =/uni222B.alt0251 49/u1D4661/u1D4531(/u1D4661)/u1D451/u1D4661 =/uni222B.alt0251 49/u1D46611 2/u1D451/u1D4661=/uni0028.alt031 4/u1D4662 1/uni0029.alt03/u1D4661=51 /u1D4661=49= 50, /u1D4532(/u1D4662) =/uni222B.alt02∞ −∞/u1D453(/u1D4661,/u1D4662)/u1D451/u1D4661 =/uni222B.alt0251 491 12/u1D451/u1D4661=1 12(/u1D4661)/u1D4661=51 /u1D4661=49=1 6, /u1D438[/u1D44C2] =/uni222B.alt028 2/u1D4662/u1D4532(/u1D4662)/u1D451/u1D4662 =/uni222B.alt028 2/u1D46621 6/u1D451/u1D4662=/uni0028.alt031 12/u1D4662 2/uni0029.alt03/u1D4662=8 /u1D4662=2= 5, then covariance is Cov(/u1D44C1,/u1D44C2) =/u1D438(/u1D44C1/u1D44C2)−/u1D438(/u1D44C1)/u1D438(/u1D44C2) = 250−(50)(5) = (choose one) (i) 0(ii)1(iii)2. ii. Since /u1D44C1and/u1D44C2are uncorrelated, Cov( /u1D44C1,/u1D44C2) = 0, then correlation is /u1D70C=Cov(/u1D44C1,/u1D44C2)/uni221A.alt01 /u1D449(/u1D44C1)/u1D449(/u1D44C2)=0√0.245×0.245= (choose one) (i) 0(ii)1(iii)2. iii. Random variables /u1D44C1and/u1D44C2are uncorrelated, Cov( /u1D44C1,/u1D44C2) = 0, be- cause they are independent , /u1D438(/u1D44C1/u1D44C2) =/u1D438(/u1D44C1)/u1D438(/u1D44C2). (i)True (ii)False Section 7. The Covariance of Two Random Variables (ATTENDANCE 9) 167 (b)Machine B. Bivariate density function for machine B is /u1D453(/u1D4661,/u1D4662) =/uni007B.alt031 4,49≤/u1D4661≤51,4≤/u1D4662≤6 0 elsewhere i. Since/u1D44C1and/u1D44C2are independent, Cov(/u1D44C1,/u1D44C2) = (choose one) (i) 0(ii)1(iii)2. ii. Since /u1D44C1and/u1D44C2are independent, correlation /u1D70C=Cov(/u1D44C1,/u1D44C2)√ /u1D449(/u1D44C1)/u1D449(/u1D44C2)= (choose one) (i) 0(ii)0.5(iii)1. 4.More Covariance and Correlation. Consider /u1D453(/u1D4661,/u1D4662) =/uni007B.alt036 5(2−/u1D4661)(1−/u1D4662) 0≤/u1D4661≤2,0≤/u1D4662≤1, /u1D4661+2/u1D4662<2 0 elsewhere which has a non–rectangular range. (a) Since /u1D438[/u1D44C1/u1D44C2] =6 5/uni222B.alt021 0/uni222B.alt022 0(/u1D4661/u1D4662)(2−/u1D4661−2/u1D4662+/u1D4661/u1D4662)/u1D451/u1D4661/u1D451/u1D4662 =6 5/uni222B.alt021 0/uni222B.alt022 0/uni0028.alt01 2/u1D4661/u1D4662−/u1D4662 1/u1D4662−2/u1D4661/u1D4662 2+/u1D4662 1/u1D4662 2/uni0029.alt01 /u1D451/u1D4661/u1D451/u1D4662 =6 5/uni222B.alt021 0/uni0028.alt03 /u1D4662 1/u1D4662−1 3/u1D4663 1/u1D4662−/u1D4662 1/u1D4662 2+1 3/u1D4662 1/u1D4662 2/uni0029.alt03/u1D4661=2 /u1D4661=0/u1D451/u1D4662 =6 5/uni222B.alt021 0/uni0028.alt03 4/u1D4662−8 3/u1D4662−4/u1D4662 2+8 3/u1D4662 2/uni0029.alt03 /u1D451/u1D4662 =6 5/uni222B.alt021 0/uni0028.alt034 3/u1D4662−4 3/u1D4662 2/uni0029.alt03 /u1D451/u1D4662 =6 5/uni0028.alt034 6/u1D4662 2−4 9/u1D4663 2/uni0029.alt03/u1D4662=1 /u1D4662=0=4 15, /u1D438[/u1D44C1] =/uni222B.alt022 0/u1D4661/u1D4531(/u1D4661)/u1D451/u1D4661=6 5/uni222B.alt022 0/u1D4661/uni0028.alt03 1−1 2/u1D4661−1 4/u1D4662 1+1 8/u1D4663 1/uni0029.alt03 /u1D451/u1D4661 =6 5/uni0028.alt031 2/u1D4662 1−1 6/u1D4663 1−1 16/u1D4664 1+1 40/u1D4665 1/uni0029.alt03/u1D4662=2 /u1D4662=0=14 25, /u1D438[/u1D44C2] =/uni222B.alt021 0/u1D4662/u1D4532(/u1D4662)/u1D451/u1D4662=6 5/uni222B.alt021 0/u1D4662/uni0028.alt01 2−2/u1D4662−2/u1D4662 2+2/u1D4663 2/uni0029.alt01 /u1D451/u1D4662 =6 5/uni0028.alt03 /u1D4662 2−2 3/u1D4663 2−2 4/u1D4664 2+2 5/u1D4665 2/uni0029.alt03/u1D4662=1 /u1D4662=0=7 25, 168 Chapter 5. Multivariate Probability Distributions (ATTENDANCE 9) then covariance is Cov(/u1D44C1,/u1D44C2) =/u1D438(/u1D44C1/u1D44C2)−/u1D438(/u1D44C1)/u1D438(/u1D44C2) =4 15−/uni0028.alt0314 25/uni0029.alt03/uni0028.alt037 25/uni0029.alt03 = (choose one) (i)205 1875(ii)206 1875(iii)207 1875. (b) Since /u1D438[/u1D44C2 1] =/uni222B.alt022 0/u1D4662 1/u1D4531(/u1D4661)/u1D451/u1D4661=6 5/uni222B.alt022 0/u1D4662 1/uni0028.alt03 1−1 2/u1D4661−1 4/u1D4662 1+1 8/u1D4663 1/uni0029.alt03 /u1D451/u1D4661 =6 5/uni0028.alt031 3/u1D4663 1−1 8/u1D4664 1−1 20/u1D4665 1+1 48/u1D4666 1/uni0029.alt03/u1D4662=2 /u1D4662=0=202 25, /u1D449[/u1D44C1] =/u1D438/uni005B.alt01 /u1D44C2 1/uni005D.alt01 −[/u1D438[/u1D44C1]]2=202 25−/uni0028.alt0314 25/uni0029.alt032 =4854 625, /u1D438[/u1D44C2 2] =/uni222B.alt021 0/u1D4662 2/u1D4532(/u1D4662)/u1D451/u1D4662=6 5/uni222B.alt021 0/u1D4662 2/uni0028.alt01 2−2/u1D4662−2/u1D4662 2+2/u1D4663 2/uni0029.alt01 /u1D451/u1D4662 =6 5/uni0028.alt032 3/u1D4663 2−2 4/u1D4664−2 5/u1D4665 2+2 6/u1D4666 2/uni0029.alt03/u1D4662=1 /u1D4662=0=3 25, /u1D449[/u1D44C2] =/u1D438/uni005B.alt01 /u1D44C2 2/uni005D.alt01 −[/u1D438[/u1D44C2]]2=3 25−/uni0028.alt037 25/uni0029.alt032 =26 625, then correlation is /u1D70C=Cov(/u1D44C1,/u1D44C2)/uni221A.alt01 /u1D449(/u1D44C1)/u1D449(/u1D44C2)=206 1875/uni221A.alt02 4854 625×26 625≈ (choose one) (i) 0.19(ii)0.23(iii)0.34.