p
A
y
x
y
x
p
A
Y
A
X
A
XY
∈
∈
∈
,
,
,
,
,
,
d)
Are X and Y conditionally independent, given that they belong to group A?
Why/why not?
e)
Find and plot
(
29
y
x
p
Y
X
.
f)
Compute and plot E(XY ). Use this to find E(X).
g)
Find
(
29
(
29
( 29
(
29
(
29
Y
E
Y
X
E
X
E


. (This expectation is known as the covariance of X and Y, cov(X,Y))
h)
If cov(X,Y)=0 then X and Y are said to be uncorrelated. Are X and Y uncorrelated?
i)
Some other joint PMFs (all uniform) are given. For each case, state whether X and Y are independent, uncorrelated?
3)
After collecting some data, the rms voltage, V, at the mains is modeled as a Gaussian (although rms values are never negative)
with mean 220 V and variance 100 V
2
.
Answer the following questions using “the standard normal table” on page 155 of the textbook.
a)
Find the probability that V < 200.
b)
Find the probability that 205 < V < 235.
c)
Find the probability that V > 250.
4)
a)
Find and plot the cumulative distribution function of an exponential random variable whose mean value is 3.
b)
Using the CDF you obtained in part(a) find the probability to have an observation in [1,3].
5)
X
is an exponential random variable with mean 4.
a)
Find and plot
( 29
( 29
A
x
F
A
x
f
A
X
A
X
,
where the event
A
is
{ }
a
X
A
≥
=
,
a
is a positive constant
b)
Find
[ ]
A
X
E
. How are
[ ]
A
X
E
and
[ ]
X
E
related to each other?
4
3
2
1
0 1
2
3
4
5
6
7
k
p
X
(
k
)
1/12
x
y
1
3
2
4
x
y
3
2
x
y
1
3
2
4
3
x
y
2
2
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '12
 Odtu
 Normal Distribution, Probability theory, East Technical University, Middle East Technical University

Click to edit the document details