NTHU MATH 2810
Final Examination
Jan 8, 2008
Note.
There are 8 problems in total. The total score is 100pts.
To ensure consideration
for partial scores, write down intermediate steps where necessary.
1. (
16pts, 2pts for each
) For the following statements, please answer true or false. If false,
please explain why.
(a) Let
X
be a continuous random variable, then
P
(
X
∈
A
) = 0 for any countable set
A
.
(b) For a continuous random variable, the values of its probability density function
(pdf) must be between 0 and 1.
(c) Transformation by using Jacobian can be applied to ﬁnd the joint pdf when the
mapping between two groups of
n
random variables is not onetoone.
(d) A random variable
X
with possible values 0 and 1 will have
E
(
X
k
) =
E
(
X
) for
k
= 2
,
3
,
4
,...
.
(e) Let
X
1
,...,X
n
be i.i.d. from a distribution with ﬁnite variance. The variance of
X
n
= (
X
1
+
···
+
X
n
)
/n
always tends to zero as the sample size
n
increases to
inﬁnity.
(f) The correlation coeﬃcient of two independent random variables is zero.
(g) If
X
and
Y
are uncorrelated, then
E
(
X

Y
) =
E
(
X
).
(h) If
X
and
Y
are independent, then
E
(
XY
) =
E
(
X
)
E
(
Y
) and
E
(
X/Y
) =
E
(
X
)
/E
(
Y
).
2. (
15pts, 3pts for each
) For each of the random variables
X
below, determine the type of
distribution (i.e., Normal, Exponential, Gamma, Beta, Uniform, Poisson, Hypergeomet