AMS 310: Survey of Probability and Statistics
MIDTERM II Spring 2009
Last Name:
First Name:
ID:
Show all your work for full credit
1. Let
X
be a continuous random variable having the probability density function
f
(
x
) =
‰
1
 
x

if

c
≤
x
≤
c
,
0
otherwise.
where
c >
0 is a constant.
(a) (2pts) Plot the function
f
(
x
).
(b) (3pts) Find
c
.
(b) (3pts) Find the variance
V ar
(
X
).
(d) (4pts) Determine the first quartile
Q
1
and the median
Q
2
of
X
.
2. Let
X
and
Y
be two continuous r.v.’s with joint density function
f
(
x, y
) =
‰
4
9
(2 +
xy
)
if 0
≤
x
≤
1, 0
≤
y
≤
1,
0
otherwise
1
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
(a) (5pts) Find the marginal distribution
f
X
(
x
).
(b) (5pts) Find
P
(
X
≤
1
2
).
(c) (5pts) Find the probability
P
(
X
≤
1
2
, Y
≤
1
2
).
(d) (5pts) Find the conditional probability
P
(
X >
1
2

Y >
1
2
).
3. Let
W
be a continuous random variable having exponential distribution with parameter
λ
= 2
.
5.
(a) (4pts) Find the mean
μ
and variance
σ
2
of
W
.
(b) (4pts) Find
P
(
W >
6
.
5

W >
2
.
5).
4. Suppose that
X
and
Y
are two discrete r.v.’s with joint mass function
f
(
x, y
) =
‰
a
(
x
+
y
)
for
x
= 0
,
1
,
2 and
y
= 1
,
2
,
3
0
otherwise
2
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '08
 Mendell
 Normal Distribution, Probability distribution, Probability theory, probability density function, Cumulative distribution function

Click to edit the document details