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# SolutionsProbabilityPrimer.pdf

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York University

Economics

ECON 3210

Razvan Sufana

Fall

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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
u▯E()▯ u ¦0.50
x
(c) Expected profit is given by
E() 5E(▯X25u▯▯)2 5102 3
(d) The variance of profit is given by
2
var(Y )Xvar(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, 44
EXERCISE P.3
2
Assume that total sales X are measured in millions of dollars. Th▯n,5,0.3▯, and
P▯▯!3 PZ ! § · 53▯2
© ¹ 0.3
PZ▯ ▯ 1.6667
7 6 6 6 . 1 1P▯ ▯
29.0
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, 0.52 0.25 .
! s t n i a S o G Probability Primer, Exercise Solutions, Principles of Econometrics, 4e
EXERCISE P.6
0 6 4 8 5 7 0 3 4 0 1 7 0 ( )0 3 4 0 1 7 0 0 ) 3 4 0 1 7 0(a)4 ES()S E ▯
(b) var(SALES) ▯▯ar(407
(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
2 1 0
C
f (b
(a) The marginal probability distribution for C is given in the last column of the above table.
(b) u ▯u EC()u
(c) ▯ u C ▯Ear(f)cu ▯
(d) For the two companies’ advertising strategies to be independent, the condition
l l a r o f d l o h t s u m
f
s i t r e v d a ’ s e i n a p m o c o w t e h t , s u h T
(e) Values for A aregivenbytheequation
obtained by matching values obtained from this equation with corresponding probabilities
for B.
(f) Since the relationship between A and B is an exact linear one, they are perfectly correlated.
The correlation between them is 1. f
f o n a e m e h t o t e s o l c y l e v i t a l e r e r a s a f o a e r a e h t r o f a l u m r o f e h t t a h t g n i Probability Primer, Exercise Solutions, Principles of Econometrics, 4e1 1
EXERCISE P.10
§ · ▯ 1 1
P P(a) ▯ EZ(P) E ▯¨ ¸ 2(>[email protected] ()Y ( )
© ¹2
(b) Assuming X and Y are independent,
2 2
V Y ▯ var(ZV Xvar§▯·Y▯ 1 > @r( ) var( ) ( 2 2 ) V
© ¹ 2 2 ▯2 4
2
(c) Assuming that cov(XY, ) V0.5 ,
§ ·Y▯ 1 2
var(Z) var© ¹ 2 ▯2 > @r(X )Y vXr( ) 2cov( , )
2
▯u▯ ▯ 2 5.0 3V
4 4 9 8 2 0 . 0 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 havA 4▯ 8 2 ▯and RB▯ ▯, 12 2▯.
(a) E()PE52.▯▯.0 ▯2.0 57RE ▯ R ▯ ▯ ▯ ▯
A B A B
▯025 4 0.75 8 7
(b) var▯▯ RV 2P ABar▯0.25 ▯ 0.75 ▯
u u u 0.25▯var ▯ ▯ 0.75 var R▯ ▯2 0.25 0.75 cov ▯ , ▯
A B B A
, w o N
U 1 cov▯RRA B ▯
var▯▯ var▯▯
A B
, e c n e H
cov(RR, ) V V 8 12 96
A B AB
2 2 2 2
var(P) ▯uuu25 8 0.75 12 2 0.25 0.75 96 121
V P 121 11
(c) When
U 0.5 cov▯RRA B ▯
var▯▯ var▯▯
A B
cov ▯RA B ▯Vu0.AB 0.5 8 12 48
var(P) ▯uuu252 28 20.75 12 2 0.25 0.75 48 103
V 103 10.15
P
(d) When ȡ 0 c,v RR▯ ▯ 0 , and the variance and standard deviation of the portfolio are
A B
2 2 2 2
var(P) ▯u0.25 8 0.75 12 85
V P 85 9.22 Probability Primer, Exercise Solutions, Principles of Econometrics, 4e

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