Econ 325
Assignment 4
Please refer to “Notes on Mathematical Expectation, Variance, and Covariance”
https://canvas.ubc.ca/courses/130366/files/30803760/download?download_frd=1
1. Let Xand Ybe two discrete random variables. The set of possible values for X
is {x1, . . . , xn}; and the set of possible values for Yis {y1, . . . , ym}. The joint pmf
(probability mass function) is given by
pX,Y
ij =P(X=xi, Y =yj), i = 1,...n;j= 1, . . . , m.
The marginal pmf of Xis
pX
i=P(X=xi) =
m
X
j=1
pX,Y
ij , i = 1, . . . n,
and the marginal pmf of Yis
pY
j=P(Y=yj) =
n
X
i=1
pX,Y
ij , i = 1, . . . n,
By definition of conditional mass function, we can express the conditional mass function
of Ygiven X=xas P(Y=yj|X=xi) = pX,Y
ij
pX
i
. Please use summation operator for
proof whenever possible. Let a,b, and cbe constant.
(a) Prove that, if Xand Yare stochastically independent, then Cov(g(X), Y ) = 0
for any function g.
(b) Let g1(x) and g2(x) be some functions of x. Prove that V ar (g1(X) + g2(X)) =
V ar (g1(X)) + V ar (g2(X)) + 2C ov (g1(X), g2(X)).
(c) Let bbe a constant. Show that E[(X−b)2] = E(X2)−2bE(X) + b2. What is the
constant value of bthat gives the minimum value of E[(X−b)2]?
(d) Define Z= (X−E(X))/pV ar(X). Prove that E[Z] = 0 and V ar[Z] = 1.
(e) Define Z= (X−E(X))/pV ar(X). Prove that Corr(X, Z ) = 1.
(f) Consider another random variable Zin addition to Xand Y. Prove that V ar(aX+
bY +cZ) = a2V ar(X) + b2V ar(Y) + c2V ar(Z) + 2abC ov(X, Y ) + 2acC ov(X, Z )+
2bcCov(Y, Z ) for any constant a,b, and c.
(g) Show that Corr(X, Y ) = −1 or 1 if Y=a+bX .
(h) Show that EX[EY[Y|X]] = EY[Y].
2. Suppose that {X1, X2, ..., Xn}is a random sample, where Xitakes a value of zero or
one with probability 1 −pand p, respectively. Define ¯
X=1
nPn
i=1 Xi. Show that
E(¯
X) = pand Var( ¯
X) = p(1 −p)
n.
1
