Chapter 2 Answer Key

Chapter 2 Answer Key

• A random variable is random because its numerical value is unknown until the

variable is realized or observed.

• A discretely distributed random variable can, with some probability, take any

one of a discrete set of values.

3.m X i=1 Pi=P1+P2+P3+:::::::::::+Pm1+Pm= 1 4.1 6 + 1 6 + 1 6 = 3 6 = 1 2

• Here there must be onlym=4 possible states, since the given probabilities add

up to one. The expected value for this discretely distributed random variable is thus P 4 j=1 cjpjwherec1= 5; c2= 8; c3= 9; c4= 11 andp1= 0:1; p2= 0:4; p3= 0:2; p4= 0:3 So = 4 X j=1 cjpj= (5)(0:1) + (8)(0:4) + (9)(0:2) + (11)(0:3) = 8:8 and V ar(Y) =E[(Y) 2 ]

= (0:1)[58:8]

2

+ (0:4)[88:8]

2

+ (0:2)[98:8]

2

+ (0:3)[118:8]

2

= 3:156

• The expected value of a random variable is a xed number that can be calcu-

lated with a given probability distribution. It will be a xed number across any set of samples taken. Whereas,

Yis the random variable whose realization is the xed value yin a particular sample.

7.

E[Y] =

1 8

[1 + 2 + 3 + 4] +

1 4

[7 + 8] = 5:0

var[Y] = 8 X i=1 pi[ci] 2

= [(15)

2

+ (25)

2

+ (35)

2

+ (45)

2

](1=8) + [(75)

2

+ (85)

2

](1=4)

= 7 1 There is no Answer Key for Chapter 1 (Fundamentals of Applied Econometrics 1e Richard A. Ashley) (Solution Manual, For Complete File, Download link at the end of this File) 1 / 3

• E[Y] is usually unknown because a probability distribution of the random vari-

able is usually unknown.

• Thek

th central moment around 0 is equal toE[(Y0) k

] =E[Y

k ]. Thek th central moment around the mean is equal toE[(Y)] k . Thek th central moment around 0 measures how any distribution of a random variable varies around the specic value of zero. The dierence between thek th central moment around the mean is a measure how the random variable Y varies around the mean.

10.E[Y 3

] = (:2)(1

3

)+(:2)(2

3

)+(:2)(3

3

)+(:1)(4

3

)+(:1)(5

3

)+(:1)(6

3

)+(:1)(7

3

) = 82

var[Y 3

] = (:2)(1

3 82) 2

+ (:2)(2

3 82) 2

+ (:2)(3

3 82) 2

+ (:1)(4

3 82) 2 +

(:1)(5

3 82) 2

+ (:1)(6

3 82) 2

+(:1)(7

3 82) 2

= 11836:6

• A realized value y can take on any real value from [1;1]

12.

E[Y3] =E[Y]3 = 11:343 = 8:34

13.

E[3Y] = 3E[Y] = (3)(25) = 75

14.

var[Y8] = 11:34

15.var[3Y] = (9)var[Y] = (9)(25) = 225 16.

E[3Y+ 7] = 3E[Y] + 7 = (3)(11) + 7 = 40

var[3Y+ 7] =var[3Y] = 9var[Y] = (9)(3) = 27:

• Letcov(Y; Z)<0. This means that X and Y have an inverse linear relationship.

Note that the covariance does not sensibly quantify the strength of the linear relationship, since the covariance depends on the units in which X and Z are measured, but it indicates that thereisa relationship and that (approximated by a linear relation) it would have a negative slope.

• The correlation between two random variables quanties the sign and strength

of a linear approximation to the relationship, if any, between them. The co- variance does the same thing with regard to the sign, but not the strength of the relationship.

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• A correlation can take on any real number between [1;1].

20.c1::::cnmust not be random, they must be xed constants. I.e they cannot depend in anyway on the random variablesY1:::::Yn

• The covariance of Y and Z isEf[YE(Y)][ZE(Z)]g

=EfY ZY EfZg ZEfYg+EfYgEfZgg =EfY Zg EfYgEfZg EfZgEfYg+EfYgEfZg =EfY Zg EfYgEfZg

= 13(4)(3) = 1

• For Y and Z to be uncorrelated, their covariance must be zero. Therefore,

E[Y Z] =E[Y]E[Z], soE[Y Z] = 12.

• They are not related at all.
• Since Y and Z are assumed to be independently distributed,p1;4in this case

equals the probability of observingYiin state 1 regardless of the value of Z, multiplied by the value ofZiin state 4 regardless of the value of Y.

• Since Y and Z are independently distributed,

E[g(Y)w(Z)] =E[g(Y)]E[w(Z)] Therefore,E[w(z)] = 12 4 = 3

• Let Y and Z be independent random variables. So thecorrf(Y; Zg= 0. It

must follow that thecovfY; Zg= 0.

• No, ifcovfY; Zg= 0 then it implies that there does not exist a linear relation-

ship, but there very well could be a non-linear relationship between the two variables.

• When two random variables, Y and Z, are identically distributed then this

means that both of the variables are realizations of `picks' from the same dis- tribution. Thus, for example, all of the population moments (mean, variance, and higher moments) of Y are equal to the corresponding population moments for Z.

• If the mean and variance of anormallydistributed variable are given, then

everything about its distribution is completely known. If it is not known that the variable is normally distributed, then its distribution is not completely known { e.g., in terms of how symmetrically the variable is distributed around its mean, how thick the tails of its distribution are, etc.

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