5609-5-122P
AID: 1825 | 06/06/2014
Construct the histogram with the range of values k from 4 to 4 by using Octave.
Octave coding:
octave-3.6.4.exe:1> U1=rand(1000,1);
octave-3.6.4.exe:2> U2=rand(1000,
5609-6-31P
AID: 1825 | 14/02/2014
(a)
Find
E Z
by integrating over
fZ z
:
From Problem 6.27,
1
2
ln z
2
fZ z
The expectation of Z is obtained below:
1
E Z z
0
1
2
ln z dz
2
1
1
2
z ln z dz
20
Us
5609-5-132P
AID: 1825 | 21/02/2014
(a)
Find the probability density function of
R2
.
Let X and Y be the Gaussian random variables with zero mean and unit variance.
Consider
R X Y
2
2
as the total ener
5609-6-34P
AID: 1825 | 21/02/2014
(a)
Find the mean vector in Problem 6.5a.
From Problem 6.5a, the probability values of X are given below:
1
8
3
P X 1
8
3
P X 2
8
1
P X 3
8
P X 0
The probability
5609-5-133P
Define
Xn
AID: 1825 | 21/02/2014
and
Yn
as follows:
The output sequence of
The output sequence of
is,
Xn
U U n1
Xn n
2
is,
Yn
Yn
U n U n 1
.
.
2
(a)
Find the joint probability density
5609-6-74P
Compute
AID: 1825 | 21/02/2014
.
VAR X 1
VAR X 1 E X 1 E X 1
2
Obtain
E X1
2
as follows:
1
E X 1 x1dx1
0
x12
2
1
0
1 2
1 0
2
1
2
Obtain
E X 12
as follows:
1
E X 12 x12 dx1
0
1
x13
5609-7-8P
AID: 1825 | 21/02/2014
(a)
Find the characteristic function of Z.
Z E ei Z
Since
Z 3 X 7Y
, the above equation becomes,
Z E ei 3 X 7Y
E e
i 3 X
e
i 7 Y
Since X and Y are independent ra
5609-6-90P
AID: 1825 | 21/02/2014
(a)
Find the coefficients of the best linear estimator (interpolator).
The orthogonality principle is given below:
E X 2 aX 2 bX 3 X 1 0
E X 2 aX 2 bX 3 X 3 0
That is
5609-7-37P
AID: 1825 | 14/02/2014
From Problem 7.36, the Chernoff bound for X, where X is a Gaussian random variable
with mean and variance
.
2
P X a e
a
(1)
2
,a
2 2
From Equation 4.54, the expre
5609-7-39P
AID: 1825 | 14/02/2014
a)
Find the Chernoff bound for the gamma distribution function with parameters
Let X be a gamma distribution function with parameters
and
and
.
.
The Chernoff bound f
5609-7-43P
AID: 1825 | 14/02/2014
(a)
Consider,
Yn 2n X 1 X 2 K X n
Where,
be a sequence of independent, identically distributed (iid), equiprobable
Xn
Bernoulli random variables.
Define
Yn
as follows
5609-7-52P
AID: 1825 | 14/02/2014
It is given that the clock does not tick with probability
1 p
and
p 0.1
.
Find the rate at which the clock moves forward.
The time between forward ticks of the clock
5609-7-54P
AID: 1825 | 21/02/2014
(a)
Consider
That is,
N t n
.
implies that
N t n
Figure (1) represents
S n t
S n t
.
.
Figure (1)
Figure (2) represents
S n t
.
Figure (2)
From Figure (1) and Fi
5609-7-80P
AID: 1825 | 21/02/2014
(a)
The Gaussian function is given below:
Sn X 1 X 2 K X n
From Problem 7.3, the mean and variance of
Sn
are given as follows:
(1)
E Sn n
VAR S n
If mean
and varian
5609-7-64P
AID: 1825 | 14/02/2014
From Problem 7.63, the mean residual life is given by:
(1)
E X 2
E R
2E X
a)
Let
X j 's
0, 2
be independent, identically distributed uniform random variables in t
5609-8-8P
AID: 1825 | 07/02/2014
(a)
Let
be a pair of random variables with known means
X ,Y
1
and
2
. The estimator
for the covariance is given below:
n
1
C X ,Y X j 1 Y j 2
n j 1
Find the expecte
5609-8-13P
AID: 1825 | 07/02/2014
(a)
Prove that the estimator
Consider
and
1
2
p1 1 p 2
is also an unbiased estimator of
as an unbiased estimator for
.
.
The estimator is given below:
p1 1 p 2
Take
5609-5-108P
AID: 1825 | 14/02/2014
Find the joint probability distribution function of W and
.
Let X and Y be independent Gaussian random variables with mean and variance 0 and 1.
Figure (1)
From Figu
5609-5-120P
AID: 1825 | 14/02/2014
(a)
Estimate the value of the signal X from the noisy observation Y as given below:
X
1
Y
2
N
1 2
X
cY
Where,
2
X
c 2
2
X N
From Example 5.47,
Y X N
Find the mean
3864-8-14E
AID: 1825 | 17/08/2013
Create a simple two-group example to show that for 30 subjects, power increases as the
sample size becomes more nearly equal.
Find the unequal sample sizes:
Test the
3864-9-37E
AID: 1825 | 07/09/2013
The data represents the trials labeled Y in the condition in which there are five digits in
the comparison set taken from the data set RxTime.dat [300 observations].
5609-4-3P
AID: 1825 | 07/02/2014
(a)
Find the cumulative distribution function of the random variable Z(t).
The point where the clock hand comes to rest is given by the coordinates (x, y); and
define
5609-2-107P
AID: 1825 | 14/02/2014
(a)
The urn has two black balls (denote it by B) and two white balls (denote this by W).
The trellis diagram in Figure (1) shows the indefinitely repeated all possib
5609-4-79P
AID: 1825 | 07/02/2014
(a)
Find d such that the probability of X lying outside the range is 1%.
That is, to find d when
P X 4d X 4d 0.01
P X 4d X 4d P X 4d P X 4d
4 d 1 4 d
1 Q 4 d 1 1
5609-4-92P
AID: 1825 | 07/02/2014
(a)
From Problem 4.18,
f X x 4x 1 x , 0 x 1
2
.
The area covered by the disc with radius x, is given by the equation given below:
y x2
Differentiate the above equatio
5609-4-93P
AID: 1825 | 28/02/2014
The quantizer defined in Example 4.20 is as follows:
Z X q X
Also, the intervals are given below:
, 3d , 3d , 2d , 2d , d , d , 0 , 0, d , d , 2d , 2d ,3d , 3d ,
T
5609-4-166P
AID: 1825 | 21/02/2014
(a)
Identify the best code to indicate which pattern of k 1s and n k 0s occurred.
For the given n outputs of a binary information source, there are k 1s and n k 0s.
5609-4-171P
AID: 1825 | 14/02/2014
Find the maximum entropy probability density function of X.
From Problem 4.170,
VAR X
2
f X x Ce
1g1 x 2 g 2 x
. Also, it is given that
E X m
and
.
Thus, the rand
5609-5-70P
AID: 1825 | 21/02/2014
(a)
Calculate the correlation coefficient of Example 5.28.
Let X be a discrete random variable with
Calculation of
9
pX 0
16
,
6
p X 1
16
.
pX 0
1 1 1
4 4 16
4 4