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JNTUH B.Tech II PROBABILITY THEORY AND STOCHASTIC PROCESSES Previous Question Paper (April/May - 2023 - R13)

B.Tech | II | II-I | R13 | April/May - 2023

Answer PaperPaper Code: 113BTMaximum Marks: 75Time: 3 Hours

Note: i) Question paper consists of Part A, Part B. ii) Part A is compulsory, which carries 25 marks. In Part A, Answer all questions. iii) In Part B, Answer any one question from each unit. Each question carries 10 marks and may have a, b as sub questions.

Part A

What are the conditions for a function to be a random variable?

2 marks

Answer

A function X(s) mapping the sample space S to the set of real numbers is a random variable if it satisfies the following two conditions: 1) The set {s : X(s) ≤ x} is an event for every real number x, meaning it must be a subset of the sample space S that belongs to the defined sigma-algebra. 2) The probabilities of the events {X = ∞} and {X = -∞} must be zero, ensuring the function takes on finite real values with probability one.

Give the expression for an arbitrary transformation of a single random variable.

3 marks

Answer

For a continuous random variable X with probability density function f_X(x), let Y = g(X) be a transformation where g(x) is a monotonic, differentiable function. The probability density function of the transformed random variable Y is given by:

f_Y(y) = f_X(x) |dx/dy| = f_X(g^-1(y)) |d/dy(g^-1(y))|

If the transformation g(x) is not monotonic and maps multiple values of x to the same value y, the density function is expressed as the sum over all roots x_i such that y = g(x_i):

f_Y(y) = Σ [f_X(x_i) / |g'(x_i)|]

Define joint density function.

2 marks

Answer

For a given continuous probability density function

f(x) = kx^2 e^-x

is a density function when

x ≥ 0

. Find the value of k.

3 marks

Answer

State central limit theorem.

2 marks

Answer

Write the properties of Gaussian density curve.

3 marks

Answer

Explain wide sense stationary random process.

2 marks

Answer

List the properties of Auto-correlation.

3 marks

Answer

What is the Relationship between Power Spectrum and Autocorrelation Function?

2 marks

Answer

If the Power spectrum density of

x(t)

Sxx(ω)

, find the PSD of

d/dt x(t)

3 marks

Answer

Part B

210 marks

State and prove Bayes theorem.

5 marks

Answer

A man wins in a gambling game if he gets two heads in five flips of a biased coin. The probability of getting a head with the coin is 0.7. i) Find the probability that man will win. Should he play this game? ii) What is the probability of winning if he wins by getting at least four heads in five flips? Should he play this new game?

5 marks

Answer

310 marks

A missile can be accidentally launched if two relays A and B both have failed. The probabilities of A and B failing are known to be 0.01 and 0.03 respectively. It is also known that B is more likely to fail (probability 0.06) if A failed. i) What is the probability of an accidental missile launch? ii) What is the probability that A will fail if B has failed? iii) Are events “A fails” and “B fails” statistically independent?

5 marks

Answer

Explain in detail about mutually exclusive events with an example.

5 marks

Answer

410 marks

Give expressions of probability density and distribution functions of Gaussian random variable and sketch them with parameters marked.

5 marks

Answer

Obtain the expression for a monotonic transformation of a single random variable.

5 marks

Answer

510 marks

Random variables X and Y have the joint density function

f(x,y) = 2/43 (x + 0.5)^2

for

0 < x < 2

and

0 < y < 3

. Find the second-order moments

m02

and

m20

5 marks

Answer

Find the moment generating function of the random variable whose moments are

mr = (r + 1)! 2^r

5 marks

Answer

610 marks

Find the probability of the event {X ≤ 5.5} for Gaussian random variable having

ax = 3

and

σx = 2

5 marks

Answer

Obtain the expression for conditional density

fX(X/B)

where event B is defined as

{ya ≤ Y ≤ yb}

5 marks

Answer

710 marks

Define and explain joint distribution function and joint density function of two random variables X and Y.

5 marks

Answer

Explain in detail about jointly Gaussian random variables.

5 marks

Answer

810 marks

The power spectral density of a stationary random process is given by

SXX(ω) = A; -k < ω < k; = 0; otherwise

. Find the auto correlation function.

5 marks

Answer

List the properties of power spectral density function.

5 marks

Answer

910 marks

For two jointly stationary random processes, the cross-correlation function is

RXY(τ) = 2e^-2τ u(τ)

. Find the two cross-spectral density function.

5 marks

Answer

X(t)=2ACos(Wct+2θ) is a random Process, where ‘θ’ is a uniform random variable, over (0,2π). Check the process for mean ergodicity.

5 marks

Answer

1010 marks

PSD of Z(t)=X(t) + Y(t)

5 marks

Answer

Cross spectral density of X(t) and Z(t).

5 marks

Answer

A random process has the power density spectrum

Sxx(w) = w^2 / (w^2 + 1)

. Find the average power in the random process.

10 marks

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Part B

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