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# Econometrics Probability Primer Solutions Econ 103

UCLA

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This 20 page Class Notes was uploaded by kodasbigmove on Thursday April 21, 2016. The Class Notes belongs to Econ 103 at University of California - Los Angeles taught by Rojas in Spring 2015. Since its upload, it has received 35 views. For similar materials see Econometrics in Economcs at University of California - Los Angeles.

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Date Created: 04/21/16

PROBABILITY PRIMER Exercise Solutions 1 Probability Primer, Exercise Solutions, Principles of Econometrics, 4e 2 EXERCISE P.1 (a) X isarandomvariablebecauseattendanceis notknownpriortotheoutdoorconcert. Before the concert, attendance is uncertain because the weather is uncertain. (b) Expected attendance is given by ▯ ▯▯E()▯ ▯ ▯x.01 (c) Expected profit is given by E() ▯(▯X 25▯▯▯▯2 510203 (d) The variance of profit is given by var(Y ▯ var(5 ▯▯X2000) 5 var( ) 25 240,000 6,000,000 0 3 . 0 0 3 2 5 . 0 5 4 t c e p x e e h t e s u a c e b t e b e h t e k a t d l u o h e l b a t g n i w o l l o f e h t n i n e v i g s Probability Primer, Exercise Solutions, Principles of Econometrics, 4e EXERCISE P.3 Assume that total sales X are measured in millions of dollars. Th▯n,,0.3▯ , and PX ▯▯ 3 PZ▯▯ ▯ 3.▯ ▯▯ ▯ ▯ 0.3 ▯ P▯ ▯ 1.6667 7 6 6 6 . ▯▯1 1P▯ ▯ ▯59.0 ▯ 0.0478 ( n o i t a i l i f f A l a c 3 Probability Primer, Exercise Solutions, Principles of Econometrics, 4e 6 EXERCISE P.5 (a) The probability that the NFC wins the 12 flip, given they have won the previous 11 flips is 0.5. Each flip is independent; so the probability of winning any flip is 0.5 irrespective of the outcomes of previous flips. (b) Because the outcomes of previous flips are independent and independent of the outcomes of future flips, the probability that the NFC will win the next two consecutive flips is 0.5 multiplied by 0.5. That is, 0.5 ▯ 0.25 . Go Saints! Probability Primer, Exercise Solutions, Principles of Econometrics, 4e EXERCISE P.6 0 6 4 8 5 7 0 3 4 0 1 7 0 4( )0 3 4 0 1 7 0 0 ) 3 4 0 1 7 0(a)4 ES()S E ▯ ▯ (b) var(SALES)▯▯v▯ar▯ (c) P▯ ▯ Probability Primer, Exercise Solutions, Principles of Econometrics, 4e EXERCISE P.7 After including the marginal probability distributions for both C and B, the table becomes B 2 1 0 f )c 0 0.05 0.05 0.05 0.15 C 1 0.05 0.20 0.15 0.40 2 0.05 0.25 0.15 0.45 f )b 0.15 0.50 0.35 (a) The marginal probability distribution for C is given in the last column of the above table. (b) ▯▯ ▯▯ EC )▯ ▯ ▯cf()1.4.4..1 ▯c 2 (c) ▯ ▯ C ▯var(fC)▯ ▯ ▯222( ) ▯▯( ) 0 0.15 1 0.40 2 0.45 (1.3) c (d) For the two companiesV advertising strategies to be independent, the condition f ,) ▯f()cf)b C B l l a r o f d l o h t s u m c and b. We find that f f (f0,0)▯.05 (0) ▯(0) 0.15 0.15 0.0225 C B Thus, the two companiesV advertising strategies are not independent. (e) Values for A aregivenbytheequation A▯5000▯1000B . Its probability distribution is obtained by matching values obtained from this equation with corresponding probabilities for B. A f )a 5000 0.15 6000 0.50 7000 0.35 (f) Since the relationship between A and B is an exact linear one, they are perfectly correlated. The correlation between them is 1. Probability Primer, Exercise Solutions, Principles of Econometrics, 4e9 EXERCISE P.8 (a) X f x) 1 16 2 16 3 16 4 16 5 16 6 16 1 11 1 (b) PX▯▯ ▯▯4 P▯ ▯▯ 4 Xr ▯5▯▯▯ 6 66 3 1 1 1 1 1 1 ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯(c)▯ ▯ ▯EX()▯ ▯▯xf()x1 2 36 6 6 6 6 6 6 5.3 The result E().3 ▯ meansthatifadieisrolledaverylargenumberoftimes,the average of all the values shown will be 3.5; it will approach 3.5 as the number of rolls increases. 1 1 1 1 1 1 ▯ ▯ ▯ ▯ ▯ ▯ ▯ (d) ▯ ▯ EX▯▯ ▯ x ▯fx2 () 1 2 2 2 3 4 5 6 .7 x 6 6 6 6 6 6 2 2 2 (e) var(XX)▯ E▯▯ ▯▯▯ ) 15.16667 3.5 2.91667 (f) The results for this part will depend on the rolls obtained by the stunent. Letthe average value after n rolls. The values obtained by one of us and their averages are: f o s e u l a v 0 2 X ▯▯2,1,5,3,4,1,5,5,2,4,2,2,4,2,4,4,3,▯,6,3 X 53.000 X103.200 X 20.200 These values are relatively close to the mean of 3.5 and are expected to become closer as the number of rolls increases. Probability Primer, Exercise Solutions, Principles of Econometrics, 40 1 EXERCISE P.9 (a) .8 f(x)=2/3 - 2/9x .6 fx .4 .2 0 0 1 2 3 x The area under the curve is equal to one. Recalling that the formula for the area of a triangle is half the base multiplied by the height, it is ▯3▯1▯ by . 2 3 (b) When x ▯12 , fx() 9 . The probability is given by the area under the triangle between 0 and 1/2. This can be calcul2 /1▯1 P▯X ▯▯ ▯. The latter probability is P▯ ▯2 ▯▯ ▯ 3 ▯▯▯▯ ▯ 0.69444 6 3222 Therefore, 25 11 P(0 ▯▯ ▯/2) ▯X▯ ▯▯▯ ▯▯2 3 1 36 36 0.30555 (c) To compute this probability we can subtract the area under the triangle between 3/4 to 3 from the area under the triangle from 1/4 to 3. Doing so yields 1 3 P▯ ▯ ▯P▯X ▯X ▯▯▯ ▯4 4▯ 3 ▯ ▯ 3 ▯ ▯▯ ▯32 2 ▯f▯f ▯ 2 4 24 4 ▯ ▯ ▯▯▯ ▯ ▯▯ 9 1▯ ▯ ▯ 2481 242 121 9 ▯▯144 16 ▯▯ 5 0.27778 18 Probability Primer, Exercise Solutions, Principles of Econometrics, 4e1 1 EXERCISE P.10 ▯ ▯ ▯ 1 1 ▯ ▯ ▯(a) ▯ EZ(▯) E▯▯▯ ▯ 2(▯2▯ ()Y ( ) ▯ ▯2 (b) Assuming X and Y are independent, 2 2 ▯ ▯ Y ▯ var(Z▯▯ ▯ ▯▯ ▯Y▯ 1 ▯ ▯r( ) var( ) ( 2 2 ) ▯ ▯ ▯ 2 2 ▯2 4 2 (c) Assuming that cov(XY, )▯▯0.5 , ▯ ▯Y▯ 1 2 var(Z)▯ var▯ ▯ 2 ▯2 ▯ ▯r(X )Y vXr( ) 2cov( , ) 2 ▯▯▯▯▯ ▯ 2 5.0 3▯ 4 4 6 7 1 3 . 0 8 5 6 8 . 0 8 n o t n e s b a s t n e d u t s f o r e b m u n e g a r y t i l i b a b o r p e h t t a h t w o h s s n o i t a Probability Primer, Exercise Solutions, Principles of Econometrics, 4e5 1 EXERCISE P.14 Expressing the returns in terms of percentages, we haAe 4▯ 8 2▯ and RB▯ 8▯ 12 2▯. (a) E()PE▯2.▯▯.0 2.0 7RE ▯ R ▯ ▯ ▯ ▯ A B A B ▯0▯25 4 0.75 8 7 (b) var▯▯ ▯R▯ P vAB 0▯25 ▯ 0.75 ▯ ▯ ▯ ▯ ▯ 0.25▯var▯ ▯R 0.75 var R▯ ▯2 0.25 0.75 cov ▯ , ▯ A B B A , w o N ▯▯ ▯ cov▯RRA B ▯ var ▯▯ var▯▯ A B , e c n e H cov(RR, ) ▯▯ ▯ ▯ ▯8 1▯2 96 A B AB 2 2 2 2 var(P) ▯▯▯25 8 0.75 12 2 0.25 0.75 96 121 ▯▯P 121 11 (c) When ▯▯ 0.5▯ cov ▯R,A B ▯ var ▯▯ var▯▯ A B cov ▯RA B ▯▯▯▯0.AB 0.5 8 12 48 var(P) ▯▯▯252 28 20.75 12 2 0.25 0.75 48 103 ▯▯ 103 10.15 P (d) When ▯ ▯ 0,cov ▯R, ▯▯ 0 , and the variance and standard deviation of the portfolio are A B 2 2 2 2 var(P) ▯▯0▯5 8 0.75 12 85 ▯▯P 85 9.22 Probability Primer, Exercise Solutions, Principles of Econometrics, 4e EXERCISE P.15 2 (a) ▯ xi▯x▯ 1 2 i▯1 4 5 . 1 7 4 ▯ ▯ ▯ ▯2 ▯▯x ▯ ▯ xx i1 i 4 (c) ▯ ▯▯i▯▯xx▯xx i▯1 4 (d) ▯ ▯▯i▯▯xxxx i▯ 4 5 . 5 2 1 3 ▯▯ ▯ ▯2▯y y ▯ ▯ yi i1 4 ▯ ▯▯i i▯y i▯1 4 ▯ xii▯▯4▯ (f) i▯14 x x▯4 i▯1 i Probability Primer, Exercise Solutions, Principles of Econometrics, 4e 7 1 EXERCISE P.16 4 (a) x1 2 3 4▯ x ▯ i i 1 3 (b) x2 3 x x ▯ i i▯2 4 (c) x11yxy2 2▯y3 3 4 4 ▯ ii i 1 4 (d) x13yxy2 4yx3 5 4 6 2 ii ▯ i 1 4 2 2 2 (e) x33yx▯4 4 ▯ ii ▯ 3 3 (f) (()1 1▯y2y 2y▯3 3y ▯ ▯ i i i 1 Probability Primer, Exercise Solutions, Principles of Econometrics, 4e EXERCISE P.17 4 (a) ((()▯▯▯a▯▯ ▯xb▯axb axb i 1 i 1 2 3 4 ▯▯4(b▯▯x▯1 2 3 4 3 (b) ▯ i ▯▯12▯3▯▯9▯ i▯1 3 ) 232 () 222 () 1(c)) 202 ()2x▯▯▯ ▯ ▯▯▯ ▯▯ ▯▯▯▯ ▯▯ x▯0 4 3 = 7 1 + 0 1 + 5 + 2 = 4 (d) ▯))f((▯▯ f ▯▯ ▯▯f ▯ f x▯2 ▯▯f 4) ff5) (6) 2 (e) ▯ f (y ▯ f)▯y() f ,)y f y x▯0 4 2 4 4 (f) ▯▯ (2▯)()1(▯▯▯ ▯ ▯x x ▯ x▯y2▯1 x ▯2 x ▯ 2 ▯▯(▯ ▯2▯▯▯)▯▯(2 ▯3▯ ▯) (2 4 6) 10 12 14 36 Probability Primer, Exercise Solutions, Principles of Econometrics, 4e EXERCISE P.18 4 ▯ ▯(a)▯ ▯▯▯ x ▯▯xx i 4( 1 2 3 4 i▯ 4 (b) ▯ ▯▯ ▯▯xx▯xx▯▯▯▯x ▯ ▯▯▯ i 1 i 1 2 3 ▯▯1▯3▯▯ ▯▯ ▯5 4 2 2 2 2 (c) ▯ ▯▯i▯▯xxx▯xx▯▯1 2▯ 3 i 1 ▯▯▯▯▯ ▯ ▯ ▯3▯ ▯ 4 8 3 4 9 5 2 9 (d) 1 ▯ xixxxx▯▯▯▯ 21 2 3 4 i1 nn nn n 2 2 2 (e) ▯ ▯ ▯▯i▯iixx ▯ i ▯ i 2 i11 11i1▯ n 2 2 2 ▯ ▯ ▯i ▯ inx i1 i1 Probability Primer, Exercise Solutions, Principles of Econometrics, 4e 0 2 EXERCISE P.19 n n n n n ▯ ▯ ▯ ▯ ▯ ▯▯i i▯yi i i ix x yx yx i ▯1 i ▯1 i ▯i1i ▯1 n n n ▯ ▯ ▯ ▯ ▯ ▯ iiy xnx ▯ ▯ ▯in i i▯1i▯1 i▯1 ▯ ▯ ▯ n y▯x▯▯nxiin 2 i▯1 n ▯ ▯ xyiixy i▯1

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