Reference no: EM133025694

3EL01 Probability Theory and Stochastic Processes

Part - I

Question . 1 (CO1 - Understand representation of random variables.)

(a) There are four balls in a box out of which two are white and two are black. If one ball is removed from the box randomly and then second ball is picked from the box then determine the probability value that the first ball removed is black and the second ball picked is white?

(b) Consider two statistically independent events, A and B. Suppose P(A+B) = 0.58, and P(A) = 0.4, then determine the value of P(B).

(c) Consider a Cauchy distributed random variable X~C(α). Determine the value of P(0<X≤α).

(d) Consider a random variable X with Laplace distribution such that the PDF fX(x)=0.5 exp(-|x|). Suppose another random variable Y is defined as Y=|X|, then determine the PDF of Y, i.e. fY(y). Also calculate the value of fY(2).
OR

(d) Consider a random variable X with exponential distribution such that the PDF fX(x)= exp(-x)u(x). Suppose another random variable Y is defined as Y=XX, then determine the PDF of Y, i.e. fY(y). Also calculate the value of fY(4).

Question . 2 (CO2 - Investigate characteristics of joint random variables.)

(a) Consider two statistically independent random variables Θ and ∈, where Θ is a uniform random variable such that Θ~U(0, 1); and ∈ is an exponential random variable with PDF f∈(∈) = 4 exp(-4∈) for ∈ ≥ 0, and f∈(∈) = 0 for ∈ < 0. Suppose another random variable is defined as r = Θ + ∈, and events A and B are defined as A={0<Θ≤0.25}, and B= {0.25<Θ≤0.5}, then determine the following

(i) the expression of a-posteriori PDF (Probability Density Function) of Θ given r, i.e. fΘ|r(Θ| r)

(b) (ii) the ratio, P(B|r=0.5) / P(A|r=0.5)
(iii) E(Θ| r=0.5)

b) Consider two correlated joint Gaussian random variables, X and Y such that their joint PDF fXY(x,y) = N(0, 4, 0, 4, 0.6). Suppose another random variable Z 2N is defined as Z= X/Y, then determine the following
(i) the expression of the PDF of Z, i.e. fZ(z)
(ii) which type of distribution the random variable Z has?
(ii) the value of the PDF of Z at Z=2, i.e. fZ(2)
OR
(b) Consider two uncorrelated joint Gaussian random variables, X~N(0, 4) and Y~N(0, 4). Suppose two other random variables are defined as Z= sqrt(XX + 2N YY) where Z ≥ 0, and W=arctan(Y/X) where |W|< π/2, then determine the following
(i) the value of the joint PDF of Z and W at Z=2 and W=π/4, i.e. fZW(2, π/4)
(ii) the value of the PDF of Z at Z=2, i.e. fZ(2)
(iii) the value of the PDF of W at W=π/4, i.e. fW(π/4)
(iv) Are Z and W statistically independent random variables?

Question 3 (CO3 - Make use of theorems related to random variables.)
(a) Consider Xi, i=1, 2, 3, ..., 32; are zero-mean, statistically independent Laplace random variables with a common PDF (Probability Density Function), 3A fXi(x)=0.25 exp(-|x|/2). Suppose another random variable is defined as Z=(X1+X2+X3+...+X32)/16, then find the expression for the PDF of Z, fZ(z) by applying the central limit theorem. Determine the values of fZ(z) at the following values of z.

(i) z=0, (ii) z=-1, (iii) z=sqrt(2)

(b) An insurance company has received a total of 1,00,000 applications for the claim of insurance this year. The company has settled 98,000 claims and rejected 2,000 applications from the received applications. Suppose a list of 100 applications is prepared randomly from the all received applications this year, and let N is the number of rejected applications in this list and if P(N≤k) ≥ 0.98 for some integer k then determine the value of the minimum k by applying Poisson's approximation.

(c) Suppose r = Θ + η, where r is the observation, Θ is the desirable value, η is unwanted Gaussian noise such that η~N (0, 1). Suppose there are following eight independent observations made for r, listed as r = {5.022, 6.389, 6.191, 3.286, 5.340, 4.389, 4.823, 4.560}, then determine the following
(i) the value of the ML (Maximum Likelihood) estimator of Θ
(ii) the value of the variance in the ML estimator of Θ
(d) A fair coin tossed 2000 times, determine the minimum possible probability value such that the head occur between 900 and 1100.
OR
(d) Consider a random variable X with the mean value of 1 and variance value of 4. Determine the minimum possible value of P(-3≤X≤5).

Part - II
Question 4 (CO4 - Understand propagation of random signals in LTI systems.)

Consider a stochastic process, X(t) = A sin(2πt + φ), where φ is a uniform random variable such that φ~U(0, 2π). Determine the following for the given stochastic process, X(t)

(i) the mean, μx(t), (ii) the ACF (Auto-Correlation Function), Rxx(t1, t2),

(iii) Whether the process X(t) is WSS (Wide Sense Stationary) or not and why?

Consider a zero-mean stochastic process, X(t) with the ACF Rxx(t1, t2) = 1-|t1-t2|, for |t1-t2|≤1, and Rxx(t1, t2) = 0 for |t1-t2|>1.

Suppose a random variable Z is obtained by integration of the given process X(t), over the time interval of (-1, 1), then determine the following for the random variable, Z

(i) the mean of Z, (ii) the variance of Z 4N

Question  5 (CO4 - Understand propagation of random signals in LTI systems.)

Consider a zero-mean, white and Gaussian Stationary stochastic process, X(t) 

(b) with the average instantaneous power of 2 units. Suppose the given process X(t) is the input of a memory-less system and the corresponding output of the memory-less system is the process Y(t), the determine the cross- correlation function of X(t) and Y(t), Rxy(t1, t2) for the following memory-less systems.

(i) Y(t) = 2X(t) + 5, (ii) Y(t) = X(t) X(t), (iii) Y(t) = X(t) X(t) X(t) / 4,

Consider a zero-mean, white and WSS (Wide Sense Stationary) stochastic process, X(t) with the average instantaneous power of 2 units. Suppose the given process X(t) is the input of an LTI (Linear Time Invariant) system and the corresponding output of the LTI system is the process Y(t), and given that the impulse response of the LTI system is h(t) = exp(-t)u(t), then determine the following for the output process Y(t)

(i) the mean, μy(t), (ii) the ACF (Auto-Correlation Function), Ryy(t1, t2),
(iii) Whether Y(t) is a white process or not and why?
(c) Consider a zero-mean, and WSS (Wide Sense Stationary) discrete-time stochastic process, X[n] with the ACF Rxx[n] having the r[0]= 1, r[1] = 2/3, r[2]
= 1/6, and r[k] = 0 for 3≤k. Suppose the process Y[n] = X[n] + X[n-1], then determine the following for the output process Y[n]
(i) the mean, μy[n], (ii) the ACF (Auto-Correlation Function), Ryy[n1, n2],
(iii) Whether X[n] and Y[n] are white processes or not and why?

OR

(c) Consider a zero-mean WSS (Wide Sense Stationary) discrete-time stochastic process, W[n] with the ACF (Auto-Correlation Function) Rww[n]=δ[n]. Suppose W[n] is filtered through a second-order AR (Auto-regression) system to obtain an output process X[n], such that
X[n] = -0.25 X[n-2] + W[n], then determine the following
(i) the expression for ρ[n], the normalized ACF of X[n]
(ii) the values of ρ[n] for |n|≤6

Attachment:- Probability Theory and Stochastic Processes.rar