ECO 230
Marianna Kudlyak
Economic Statistics
Spring 2007
Homework #6
Due date: Tuesday, March 6
th
Total points: 10. Graded exercises: Ex.4 (2 points), Ex.6 (2 points), Ex. 8 (3 points), and
0.7 points were subtracted for the absence of each of the remaining 3 exercises.
Bonus points: Ex. 1 (5 points)
Updated: March 1
st
, 2007
The HW contains 8 exercises; however, you are required to submit only 7 exercises:
from Ex.2 to Ex.8. Exercise 1 is optional and is not to be submitted.
Exercise 1 (Bonus). Solution will be posted later.
Exercise 2.
Show the following results, justifying each step in details:
a)
For any random variables X and Y and constants a, b, c, d, show that COV(aX +
b, cY + d) = acCOV(X,Y)
COV(aX + b, cY + d) = E[(aX + b – E(aX + b))( cY + d – E(cY + d))] = E[(aX + b –
aE(X)  b)( cY + d – cE(Y)  d)] = E[(aX – aE(X))( cY – cE(Y)] = E[ac(X – E(X))( Y
– E(Y)] = ac E[(X – E(X))( Y – E(Y)] = acCOV(X,Y)
b)
var(X  Y) = var(X) + var(Y) – 2cov(X,Y). Hint: Start from the definition of
variance.
var(X  Y) = E[(X – Y  E(X – Y))
2
] = E[(X – Y  E(X) + E(Y))
2
] = E[{(X – E(X)) –
(Y  E(Y))}
2
] = E[(X – E(X))
2
+ (Y  E(Y))
2
 2(X – E(X)) (Y  E(Y))] = E[(X –
E(X))
2
]+ E[(Y  E(Y))
2
] 2E[(X – E(X)) (Y  E(Y))] = var(X) + var(Y) – 2cov(X,Y).
Last step follows from the definition of var and cov.
Exercise 3.
Let X and Y be a pair of jointly distributed discrete random variables.
Develop a numerical example (different from the one in the textbook) to demonstrate
that if covariance between X and Y is 0, it does not imply that X and Y are
statistically independent.
Answers vary. Please see ask instructor or TA if you have question.
Exercise 4.
Newbold, Carlson and Thorne, Problem 5.84 p.173.
a. Here you need to find marginal distribution function of Y
Py(0) = .08 + .03 + .01 = .12
Py(1) = .13 + .08 + .03 = .24
Py(2) = .09 + .08 + .06 = .23
Py(3) = .06 + .09 + .08 = .23
Py(4) = .03 + .07 + .08 = .18
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b. Conditional probability function:
P
YX
(y3) = 1/26; 3/26; 6/26; 8/26; 8/26 for y = 0,1,2,3,4 respectively.
c. No, because Px,y(3,4) = .08
≠
.0468 = Px(3)Py(4)
In addition answer the following questions:
d) Find the probability that a randomly chosen person from this group does not make any
purchases in a given week.
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 Spring '08
 Kudlyak
 Probability theory, TA, x,y, Joint probability distribution

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