IGNOU MMT-008 Solved Assignment 2023 | M.Sc. MACS Abstract Classes ®

Solved By – Narendra Kr. Sharma – M.Sc (Mathematics Honors) – Delhi University

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IGNOU MMT-008 Assignment Question Paper 2023

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State whether the following statements are True or False. Justify your answer with a short proof or a counter example:

a) If

P

is a transition matrix of a Markov Chain, then all the rows of

lim_{n \to \infty} P^{n}

are identical.

b) In a variance-covariance matrix all elements are always positive.

c) If

X_{1}, X_{2}, X_{3}

are iid from

N_{2} (μ, Σ)

, then

\frac{X_{1} + X_{2} + X_{3}}{3}

follows

N_{2} (μ, \frac{1}{3} Σ)

d) The partial correlation coefficients and multiple correlation coefficients lie between -1 and 1.

e) For a renewal function

M_{t}, lim_{t \to 0} \frac{M_{t}}{t} = \frac{1}{μ}

a) Let $(X, Y)$ have the joint p.d.f. given by:

f (x, y) = {\begin{cases} 1, & if | y | < x; 0 < x < 1 \\ 0, & otherwise \end{cases}

i) Find the marginal p.d.f.’s of

X

and

Y

ii) Test the independence of

X

and

Y

iii) Find the conditional distribution of

X

given

Y = y

iv) Compute

E (X ∣ Y = y)

and

E (Y ∣ X = x)

b) Let the joint probability density function of two discrete random

X

and

Y

be given as:

		$X$
		2	3	4	5
$Y$	0	0	0.03	0	0
	1	0.34	0.30	0.16	0
	2	0	0	0.03	0.14

i) Find the marginal distribution of

X

and

Y

. ii) Find the conditional distribution of

X

given

Y = 1

iii) Test the independence of variable

s X

and

Y

iv) Find

V [\frac{Y}{X} = x]

a) Let $X \sim N_{3} (μ, Σ)$ , where $μ = [5, 3, 4]^{'}$ and

\sum = (\begin{array}{ccc} 2 & 1 & 1 \\ 1 & 1 & 0.5 \\ 1 & 0.5 & 1 \end{array})

Find the distribution of:

(\begin{matrix} 2 X_{1} + X_{2} - X_{3} \\ X_{1} + X_{2} + X_{3} \end{matrix})

b) Determine the principal components

Y_{1}, Y_{2}

and

Y_{3}

for the covariance matrix:

\sum = (\begin{array}{ccc} 1 & - 2 & 0 \\ - 2 & 5 & 0 \\ 0 & 0 & 1 \end{array})

Also calculate the proportion of total population variance for the first principal component.

a) Consider a Markov chain with transition probability matrix:

P = (\begin{array}{cccc} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 \\ \frac{1}{4} & \frac{1}{8} & \frac{1}{8} & \frac{1}{2} \end{array})

i) Whether the chain is irreducible? If irreducible classify the states of a Markov chain i.e., recurrent, transient, periodic and mean recurrence time.

ii) Find the limiting probability vector.

b) At a certain filling station, customers arrive in a Poisson process with an average time of 12 per hour. The time interval between service follows exponential distribution and as such the mean time taken to service to a unit is 2 minutes. Evaluate:

i) Probability that there is no customer at the counter.

ii) Probability that there are more than two customers at the counter.

iii) Average number of customers in a queue waiting for service.

iv) Expected waiting time of a customer in the system.

v) Probability that a customer wait for 0.11 minutes in a queue.

a) A service station has 5 mechanics each of whom can service a scooter in 2 hours on the average. The scooters are registered at a single counter and then sent for servicing to different mechanics. Scooters arrive at a service station at an average rate of 2 scooters per hour. Assuming that the scooter arrivals are Poisson and service times are exponentially distributed, determine:

i) Identify the model.

ii) The probability that the system shall be idle.

iii) The probability that there shall be 3 scooters in the service centre.

iv) The expected number of scooters waiting in a queue.

v) The expected number of scooters in the service centre.

vi) The average waiting time in a queue.

b) A random sample of 12 factories was conducted for the pairs of observations on sales

(x_{1})

and demands

(x_{2})

and the following information was obtained:

\sum X = 96, \sum Y = 72, \sum X^{2} = 780, \sum Y^{2} = 480, \sum X Y = 588

The expected mean vector and variance covariance matrix for the factories in the population are:

\begin{matrix} μ = [\begin{array}{l} 9 \\ 7 \end{array}] \\ and \sum = [\begin{array}{ll} 13 & 9 \\ 9 & 7 \end{array}] . \end{matrix}

Test whether the sample confirms its truthness of mean vector at

5 %

level of significance, if:

Σ

is known,

ii)

Σ

is unknown.

[You may use:

χ_{2, 0.05}^{2} = 10.60, χ_{3, 0.05}^{2} = 12.83, χ_{4, 0.05}^{2} = 14.89, F_{2, 10, 0.05} = 4.10

]

a) Let the random vector $X^{'} = (X_{1}, X_{2}, X_{3})$ has mean vector $[- 2, 3, 4]$ and variance covariance matrix $= (\begin{array}{lll} 1 & 1 & 1 \\ 1 & 2 & 3 \\ 1 & 3 & 9 \end{array})$ . Fit the equation $Y = b_{0} + b_{1} X + b_{2} X_{2}$ . Also obtain the multiple correlation coefficient between $X_{3}$ and $[X_{1}, X_{2}]$ .

b) Define ultimate extinction in a branching process. Let

p_{k} = {b c}^{k - 1}, k = 1, 2, \dots

;

0 < b < c < b + c < 1

and

p_{0} = 1 - \sum_{k = 1}^{\infty} p_{k}

. Then discuss the probability of extinction in different cases for

E (X_{1}) \geq 1

E (X_{1}) < 1

a) If the random vector $Z$ be $N_{4} (μ, Σ)$ , where:

μ = [\begin{matrix} 1 \\ 2 \\ 5 \\ - 2 \end{matrix}]

and

\sum = [\begin{array}{cccc} 3 & 3 & 0 & 9 \\ 3 & 2 & - 1 & 1 \\ 0 & - 1 & 6 & - 3 \\ 9 & 1 & - 3 & 7 \end{array}]

Find

r_{34}, r_{34.21}

b) Suppose life times

X_{1}, X_{2}, \dots .

. are i.i.d. uniformly distributed on

(0, 3)

and

C_{1} = 2

and

C_{2} = 8

. Find:

μ^{T}

ii)

T

which minimizes

C (T)

and which is the better policy in the long-run in terms of cost.

a) Consider the Markov chain with three states, $S = {1, 2, 3}$ following the transition matrix

p = 2 [\begin{array}{ccc} 1 & 2 & 3 \\ \frac{1}{2} & \frac{1}{4} & \frac{1}{4} \\ \frac{1}{3} & 0 & \frac{2}{3} \\ \frac{1}{2} & \frac{1}{2} & 0 \end{array}]

i) Draw the state transition diagram for this chain.

ii) If

P (X_{1} = 1) = P (X_{1} = 2) = \frac{1}{4}

, then find

P (X_{1} = 3, X_{2} = 2, X_{3} = 1)

iii) Check whether the chain is irreducible and a periodic.

iv) Find the stationary distribution for the chain.

b) If

N_{1} (t), N_{2} (t)

are two independent Poisson process with parameters

λ_{1}

and

λ_{2}

respectively, then show that

P (N_{1} (t) = k [N_{1} (t) + N_{2} (t) = n] =^{n} C_{k} p^{k} q^{n - k}, where p = \frac{λ_{1}}{λ_{1} + λ_{2}}, q = \frac{λ_{2}}{λ_{1} + λ_{2}} .

a) Let $X = [\begin{array}{l} X_{1} \\ X_{2} \end{array}]$ be a normal random vector with the mean vector $μ = [\begin{array}{l} 0 \\ 1 \end{array}]$ and covariance matrix $[\begin{array}{cc} 1 & - 1 \\ - 1 & 2 \end{array}]$ . Suppose $Y = A X + b$ , where

A = [\begin{array}{ll} 1 & 2 \\ 2 & 1 \\ 1 & 1 \end{array}], b = [\begin{array}{l} 0 \\ 1 \\ 2 \end{array}] and Y \sim N_{3}

i) Find

P (0 \leq X_{2} \leq 1)

.
ii) Compute

E (Y)

.
iii) Find the covariance matrix of

Y

.
iv) Find

P (Y_{3} \leq 4)

b) A box contains two coins: a regular coin and one fake two-headed coin. One coin is chosen at random and tossed twice. The following events are defined:
A: first coin toss results in a head.
B: second coin toss results in a head.
C: coin 1 (regular) has been selected.

Find

P (A ∣ C), P (B ∣ C), P (A \cap B) ∣ C), P (A), P (B)

and

P (A \cap B)

a) Consider three random variables $X_{1}, X_{2}, X_{3}$ having the covariance matrix

[\begin{array}{ccc} 1 & 0.12 & 0.08 \\ 0.12 & 1 & 0.06 \\ 0.08 & 0.06 & 1 \end{array}] .

Write the factor model, if number of variables and number of factors are 3 and 1 respectively.

b) A particular component in a machine is replaced instantaneously on failure. The successive component lifetimes are uniformly distributed over the interval [2,5] years. Further, planned replacements take place every 3 years.

Compute

i) long-terms rate of replacements.

ii) long-terms rate of failures.

MMT-008 Sample Solution 2023

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Question:-01

State whether the following statements are True or False. Justify your answer with a short proof or a counter example:
a) If $P$ is a transition matrix of a Markov Chain, then all the rows of $lim_{n \to \infty} P^{n}$ are identical.

Answer:

The statement “If

P

is a transition matrix of a Markov Chain, then all the rows of

lim_{n \to \infty} P^{n}

are identical” is generally not true for all Markov Chains . It is true under certain conditions, such as if the Markov Chain is irreducible, aperiodic, and positive recurrent (i.e., it is ergodic).

Counterexample:

Consider a simple Markov Chain with two states

A

and

B

and the following transition matrix:

P = (\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix})

In this case, the Markov Chain is not irreducible (it consists of two disconnected states). The limit

lim_{n \to \infty} P^{n}

exists and is:

lim_{n \to \infty} P^{n} = (\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix})

Here, the rows are not identical, contradicting the statement.

Conditions for the Statement to be True:

For an ergodic Markov Chain, the statement is true. In such a case, the Markov Chain has a unique stationary distribution

π

, and:

lim_{n \to \infty} P^{n} = (\begin{matrix} π \\ π \\ ⋮ \\ π \end{matrix})

Here, all rows are identical and equal to the stationary distribution

π

So, the statement is not universally true for all Markov Chains but holds under specific conditions.

Page Break

b) In a variance-covariance matrix all elements are always positive.

Answer:

The statement “In a variance-covariance matrix all elements are always positive” is false .

Counterexample:

Consider a simple dataset with two variables

X

and

Y

, where

X = [1, 2, 3]

and

Y = [3, 2, 1]

The variance-covariance matrix for this dataset would be:

(\begin{matrix} Var (X) & Cov (X, Y) \\ Cov (Y, X) & Var (Y) \end{matrix}) = (\begin{matrix} 1 & - 1 \\ - 1 & 1 \end{matrix})

Here, the covariance between

X

and

Y

- 1

, which is not positive. Therefore, the statement is false.

Additional Notes:

The diagonal elements of a variance-covariance matrix, which represent variances, are always non-negative because variance cannot be negative.
Off-diagonal elements, which represent covariances, can be negative, zero, or positive, depending on the relationship between the variables involved.

Page Break

c) If

X_{1}, X_{2}, X_{3}

are iid from

N_{2} (μ, Σ)

, then

\frac{X_{1} + X_{2} + X_{3}}{3}

follows

N_{2} (μ, \frac{1}{3} Σ)

Answer:

The statement “If

X_{1}, X_{2}, X_{3}

are iid from

N_{2} (μ, Σ)

, then

\frac{X_{1} + X_{2} + X_{3}}{3}

follows

N_{2} (μ, \frac{1}{3} Σ)

” is true .

Justification:

Mean : The mean of $\frac{X_{1} + X_{2} + X_{3}}{3}$ is $\frac{μ + μ + μ}{3} = μ$ .
Covariance Matrix : The covariance matrix of $X_{1} + X_{2} + X_{3}$ is $Σ + Σ + Σ = 3 Σ$ because $X_{1}, X_{2}, X_{3}$ are independent. Therefore, the covariance matrix of $\frac{X_{1} + X_{2} + X_{3}}{3}$ is $\frac{1}{3^{2}} (3 Σ) = \frac{1}{3} Σ$ .

Since both the mean and the covariance matrix match the given distribution

N_{2} (μ, \frac{1}{3} Σ)

, the statement is true.

Page Break

d) The partial correlation coefficients and multiple correlation coefficients lie between -1 and 1.

Answer:

The statement “The partial correlation coefficients and multiple correlation coefficients lie between -1 and 1” is true .

Justification:

Partial Correlation Coefficients : The partial correlation coefficient measures the strength and direction of the relationship between two variables while controlling for the effect of one or more other variables. Mathematically, it is defined in a way that ensures its value lies between -1 and 1, inclusive. Specifically, it is computed as the correlation between the residuals resulting from the linear regression of each variable against the control variables. Since residuals are uncorrelated with the predicted values, the partial correlation coefficient is constrained to be between -1 and 1.
Multiple Correlation Coefficients : The multiple correlation coefficient $R$ is defined as the square root of the coefficient of determination $R^{2}$ , which is the proportion of the variance in the dependent variable that is predictable from the independent variables in a multiple regression model. Since $R^{2}$ is between 0 and 1, $R$ must be between 0 and 1. However, $R$ can be negative if the model includes a constant term and the slope is negative, but its absolute value will still be between 0 and 1.

Therefore, both the partial correlation coefficients and multiple correlation coefficients are bounded between -1 and 1, making the statement true.

Page Break

e) For a renewal function

M_{t}, lim_{t \to 0} \frac{M_{t}}{t} = \frac{1}{μ}

Answer:

The statement “For a renewal function

M_{t}, lim_{t \to 0} \frac{M_{t}}{t} = \frac{1}{μ}

” is generally true under certain conditions .

Justification:

A renewal function

M_{t}

is defined as the expected number of renewals (or arrivals, or events) that have occurred by time

t

. Mathematically, it is often defined as:

M_{t} = E [N (t)]

where

N (t)

is the number of renewals by time

t

The mean inter-arrival time (or mean time between renewals) is denoted by

μ

and is defined as:

μ = E [X]

where

X

is the random variable representing the time between renewals.

Under the assumption that

μ < \infty

and the distribution of inter-arrival times has a finite variance, it is generally true in renewal theory that:

lim_{t \to 0} \frac{M_{t}}{t} = \frac{1}{μ}

This result is often derived using advanced methods in renewal theory and stochastic processes, and it provides a way to relate the renewal function to the mean inter-arrival time

μ

So, the statement is true under the conditions that

μ < \infty

and the distribution of inter-arrival times has a finite variance.

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