Unit 3 Question bank
1.
Define
a random variable. Specify the sample space and the random variable for a coin
tossing experiment (Dec 2012)
A function which takes on any value from
the sample space and its range is some set of real numbers is called a random
variable of an experiment. For example, in a coin tossing experiment
Sample space = { H, T} H- Head, T- Tail
Random variable = {1,-1}i.e H=1 and
tail=-1
2.
When
a random process is called deterministic? (Dec
2011)
When future values of any sample
function can be predicted from knowledge of past values then the random process
is called deterministic. For example: In a die throwing random experiment, if
the future value of the throw is predicted from the past values then it is
called deterministic.
3.
When
the random process is said to be strict sense or strictly stationary? (May
2011)
The strictly stationary random process
possesses following characteristics:
The statistical properties do not change
with shift of time origin
A truly stationary process should start
at minus infinity and end at infinity
4.
Write
the Rayleigh and Rician probability density functions. (May 2011)
The Rayleigh distribution,
named for William Strutt, Lord
Rayleigh, is the distribution of the
magnitude of a two-dimensional random vector whose coordinates are independent,
identically distributed, mean 0 normal variables. The distribution has a number
of applications in settings where magnitudes of normal variables are important
Rayleigh:
Rician:
5. Differentiate
between random variable and random process
|
S.No
|
Random
Variable
|
Random
Process
|
|
1
|
It
is a set of numbers
|
It
is a wave form
|
|
2
|
It
need not be a function of time
|
It
is strictly function of time
|
|
3
|
These
are not further classified
|
It
can be stationary or ergodic
|
|
4
|
Only
ensemble averages can be calculated
|
Ensemble
as well as time averages can be calculated
|
6. Define
cumulative distribution function
The CDF of a random
variable X is the probability that a random variable X takes a value less than
or equal to x.
The variable F is
cumulative distribution and P is probability.
7. Give
the properties of CDF
1.Every cumulative
distribution function F is non-decreasing
and right-continuous,
which makes it a cad lag
function.
2. if X is a
purely discrete random variable,
then it attains values x1, x2, ... with
probability pi = P(xi), and the CDF of X
will be discontinuous at the points xi and constant in
between:
3. If the CDF F
of a real valued random variable X is continuous,
then X is a continuous random variable;
if furthermore F is absolutely continuous,
then there exists a Lebesgue-integrable
function f(x) such that
8. Define
probability density function
The derivative of CDF
with respect to some dummy variable is called PDF
The small f refers to
density function and capital F refers to distribution function.
9. Give
the properties of PDF
The properties of
probability distribution functions are
F(b) is cumulative
distribution function
f(x) is probability
density function with a dummy variable x
f(t) is probability
density function with respect to time
10. Define
mean, auto correlation and cross correlation
The mean of random
process is denoted by Mx(t). The mean
value is basically expected value of random process X(t). Autocorrelation
is the similarity between observations as a function of the time lag between
them. It is a mathematical tool for finding repeating patterns. Cross-correlation
is a measure of similarity of two waveforms
as a function of a time-lag applied to one of them. This is also known as a sliding
dot product
or sliding inner-product.
16 marks
1.
(i) Explain continuous and discrete random variables with properties (4)
(ii) Explain different
types of random process in detail (4) Dec
2010, May 2012
(iii) State and prove any 4
properties of auto correlation function (4)
(iv) Prove any 4 properties of cross correlation
function (4)
2.
(i) A three digit message is transmitted over a noisy channel having a
probability of error as P(E)=2/5 per digit. Find out the corresponding CDF (3)
(ii)Two dice are thrown at random several times. The
random variable X assigns the sum of numbers appearing on dice to each outcome.
Find the CDF for the random variable. (4)
(iii) Two
dice are thrown at random several times. The random variable X assigns the sum
of numbers appearing on dice to each outcome. Find the PDF for the random
variable. (5)
(iv) Let
X have the uniform distribution given by fx(X) = {1/2π, 0≤x≤2π
{0, elsewhere
Calculate mean, variance and standard deviation (4) (May
14)
3. (i)
Given a random process X(t) = A cos(wt+θ), where A and w are constants and θ is
a uniform random variable, show that
X(t) is ergodic in both mean and autocorrelation. (6)
May 2010
(ii) Define central limit theorem with
properties (6) May
2011
(iii) Define Gaussian process and explain its
properties in detail (4) May 2011
4. (i) In a
random process, bring out the five main properties of power spectral density (4)
May2011
(ii) Explain transmission of random
process through LTI system (4)
(iii) Define Ergodic process with ensemble
averages and time averages and explain ergodicity in mean, variance and
autocorrelation (4)
(iv) Evolve Ergodic process is always
stationary but converse is not true (4)
5. A)
A PDF of a random variable X is given by
fX
(x) = { k(1-x2), 0≤x≤1
{0, otherwise then find i) k
ii)CDF iii) P(0≤X≤2) 8marks
B) A random variable X has
PDF:
fX
(x) = {6x-6x2, 0≤x≤1
{ 0, otherwise.
Find
a and b such that P(X≤b) = P(X≥b) where
b is between 0 and 1 8 marks
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