(Statistics for Economics: 2021.12.13) Final Exam: total 45 pts
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1
1. (34 pts.) Think of two random variables (X, Y ) whose joint probability
is defined as below.
Y =0
Y =1
Y =2
X=0
1/4
1/4
0
X=1
0
1/4
1/4
(a) (2pts.) Determine the marginal probability mass function of X.
(b) (2pts.) Determine the marginal probability mass function of Y .
(c) (2pts.) Are two random variables X and Y are independent?
Explain it.
(d) (2pts.) Determine the covariance Cov(X, Y ).
(e) (4pts.) Let W = |X − Y |3 . Determine E(W ).
(f) (4pts.) Let S = X − Y and W = X + Y . Determine the joint
probability mass function pSW (s, w) = Pr(S = s, W = w) for all
(s, w).
2
(g) (4pts.) Determine E[S|W = 1] where S and W are defined in the
previous subquestion.
(h) (4pts.) Determine V ar[W |S = 0] where S and W are defined in
the previous subquestion.
(i) (6 pts.) Suppose X1 , X2 , . . . , Xn are i.i.d. random sample, and
each random variable Xi follows the probability mass function in
∑
the subquestion (a). Suppose X = n1 ni=1 Xi = 0.55. You are
interested in the parameter θ = E[Xi2 ]. (i) Suggest a consistent
estimator of the parameter θ. (ii) By using your estimator in (i),
provide the estimated value of the parameter θ. Briefly explain
your answers.
(j) (4 pts.) (continued from the previous subquestion) (4 pts.) Let
µ = E[Xi ]. Do the hypothesis testing at the 1% significance level.
H0 : µ = 0.51 and H1 : µ ̸= 0.51. Let n = 400 and X =
∑n
1
i=1 Xi = 0.55. Assume the sample size n = 400 is sufficiently
n
large.
3
2. (11pts.) Suppose
the joint probability density function of (X, Y ) is
{
cxy if 0 < x < 2, 0 < y < 1
fXY (x, y) =
0
otherwise
(a) (3pts.) Determine c so that fXY (x, y) can be a joint pdf of (X, Y ).
(b) (4pts.) Determine the marginal pdf of X and Y , fX (x) and fY (y),
respectively.
(c) (4pts.) Are two random variables X and Y independent? Briefly
explian it.
4
5
0
