IGNOU MMT 8 SOLVED ASSIGNMENT

Included:

MMT 008 (January 2025 - July 2025) - ENGLISH

Assignment

(To be done after studying all the blocks)

Course Code: MMT-008

Assignment Code: MMT-008/TMA/2025

Maximum Marks: 100

1. a) A study about a population showed that the mobility of population of a state to a village, town and city is in the following percentages:

From	To
	Village	Town	City
Village	60	25	15
Town	10	70	20
City	10	30	60

 What will be the proportion of population village, town and city after one year and two years, given that the present proportion of the population in the village, town and city are respectively 0.50, 0.40 and 0.10?

b) In a certain state, 58 landfills are classified according to their concentration of three hazardous chemicals: arsenic, barium and mercury. Suppose that the concentration of each one of the three chemicals is characterized as either high or low. If a landfill is chosen at random from among 58 landfills, given the following configuration:

	Barium
	High Mercury		Low Mercury
Arsenic	High	Low	High	Low
High	1	3	5	9
Low	4	8	10	18

 Find the probability that it has:

i) high concentration of Barium,

ii) high concentration of mercury and low concentration of both Arsenic and Barium and

iii) high concentratin of any one of the chemicals and low concentration of the other two.

2. a) Consider the following system considering of two switches I and II between two points A and B:

A signal is sent from the point A to point B and is received at B if both the switches I and II are closed. It is assumed that the probabilities of I and II being closed are 0.8 and 0.6 respectively and that P [II is closed | I is closed] = P [II is closed].

Find:

i) The probability that signal is received at B.

ii) The conditional probability that switch I was open, given that the signal was not received at B,

iii) The conditional probability that switch II was open, given that the signal was not received at B.

b) For the (M/M/K) : (∞ / FIF )0 queuing model with arrival rate λ and service rate per service channel µ, obtain the steady-state probability of n customers in the system, P . n Hence or otherwise, find the probability that an arrival has to wait.

3. a) The certain item is manufactured by three factories, say 1, 2 and 3. It is known that 1 turns out twice as many items as 2 and that 2 and 3 turn out the same number of items (during a specified production period). It is also known that 2 percent of the items produced by 1 and 2 are defective while 4 percent of those manufactured by 3 are defective. All the items produced are put into one stock pile and then one item is chosen at random. The chosen item was found defective. What is the probability that it was produced in factory 1?

b) The joint distribution of the random variables X and Y is given by:

x /y	−1	0	1
−1	α	β	α
0	β	0	β
1	α	β	α

where α,β > 0 with α + β = 4/1 .

i) Derive the marginal distribution of X and Y.

ii) Calculate the E(X), E(Y) and E(XY).

iii) Show that Cov(X,Y) = 0.

iv) Show that the variables X and Y are dependent.

	1	2	3	4
1	0	0	1	0
2	1	0	0	0
3	1/2	1/2	0	0
4	1/3	1/3	1/3	0

i) Find the probabilit 

ii) Classify the states of the given Markov chain.

b) It is claimed that the functionis the joint  distribution function of the random variables X and Y. Then:

i) determine the corresponding joint probability density functiond F_X Y,  and

ii) calculate the probability 

5. a) In an investigation related to a specific type of scores of men and women aged 65 to 70, the mean verbal and performance scores for 101 subjects were found to be:

The sample covariance matrix of the scores was

In order to test the null hypothesis that observations came from a population with mean

apply a suitable test statistic. You may consider α = 01.0 for the test.

b) What is the purpose of principal component analysis? Given the covariance matrix of order 2× ,2 explain how would you extract the principal components. Also, explain how would you find the proportion of total population variance for all the principal components.

6. a) Distinguish between ‘Age Replacement’ and ‘Block Replacement’ policies

Let the lifetimes Y₁ ,Y₂ ,....,  are independently and identically distributed random variables and follows negative exponential distribution with parameter 5. If lifetimes T > 0 and age replacement policy is to be employed, then:

i) find the mean renewal time and

ii) find the long-run average cost per unit time, given the costs C₁ 4  = and C₂ = 6units

b) In order to fit the regression line on a data set consisting of 34 pairs of values (z,y), the least square estimates of β₁ and β₂ are to be computed. From the data set, the following values are obtained:

Obtain the fitted regression line.

7. a) What is branching process? Give two real examples of branching process.

If )s(P and P )s( n respectively is the probability generating function (pgf ) of the i.i.d.

random variables and the random variables

Then show that:and

b) A community has two police cars, which operate independently of one another. The probability that a specific car will be available when needed is 0.99.

i) What is the probability that neither car is available when needed?

ii) What is the probability that a car is available when needed?

8. State whether the following statements are true or false. Justify your answer with a short proof or a counter example:

i) Although in one-step transition probability matrix, ,P of a Markov chain the sum of each row must necessarily be unity but in higher order transition probability matrices, . This rule is not necessary.

ii) For the joint pdf of random variables (X,Y) given by:

iii) Let the r.v. X follows the negative exponential distribution with parameterThen

iv) One of the examples of non-Poisson queuing system is the (M / G (:)1/ ∞ (FIF0) queuing model.

v) be a random sample from   then maximum likelihood estimators o

MMT 8 2025 (JANUARY) - English

Course Code: MMT-008

Assignment Code: MMT-008/TMA/2025

Maximum Marks: 100

1. a) A study about a population showed that the mobility of population of a state to a village, town and city is in the following percentages:

From	To
	Village	Town	City
Village	60	25	15
Town	10	70	20
City	10	30	60

What will be the proportion of population village, town and city after one year and two years, given that the present proportion of the population in the village, town and city are respectively 0.50, 0.40 and 0.10?

b) In a certain state, 58 landfills are classified according to their concentration of three hazardous chemicals: arsenic, barium and mercury. Suppose that the concentration of each one of the three chemicals is characterized as either high or low. If a landfill is chosen at random from among 58 landfills, given the following configuration:

	Barium
	High Mercury		Low Mercury
Arsenic	High	Low	High	Low
High	1	3	5	9
Low	4	8	10	18

Find the probability that it has:

i) high concentration of Barium,

ii) high concentration of mercury and low concentration of both Arsenic and Barium and

iii) high concentratin of any one of the chemicals and low concentration of the other two.

2. a) Consider the following system considering of two switches I and II between two points A and B:

A signal is sent from the point A to point B and is received at B if both the switches I and II are closed. It is assumed that the probabilities of I and II being closed are 0.8 and 0.6 respectively and that P [II is closed] I is closed] = P [II is closed].

Find:

i) The probability that signal is received at B.

ii) The conditional probability that switch I was open, given that the signal was not received at B,

iii) The conditional probability that switch II was open, given that the signal was not received at B.

b) For the (M/M/K) : (∞ / FIF )0 queuing model with arrival rate λ and service rate per service channel µ, obtain the steady-state probability of n customers in the system, P . n Hence or otherwise, find the probability that an arrival has to wait.

3. a) The certain item is manufactured by three factories, say 1, 2 and 3. It is known that 1 turns out twice as many items as 2 and that 2 and 3 turn out the same number of items (during a specified production period). It is also known that 2 percent of the items produced by 1 and 2 are defective while 4 percent of those manufactured by 3 are defective. All the items produced are put into one stock pile and then one item is chosen at random. The chosen item was found defective. What is the probability that it was produced in factory 1?

b) The joint distribution of the random variables X and Y is given by:

y/x	-1	0	1
-1	α	β	α
0	β	α	β
1	α	β	α

where α,β > 0 with α + β = 4/1 .

i) Derive the marginal distribution of X and Y.

ii) Calculate the E(X), E(Y) and E(XY).

iii) Show that Cov(X,Y) = 0.

iv) Show that the variables X and Y are dependent.

4. a) Let {X_n; }0 n ≥ be a Markov chain with four states; 1, 2, 3, 4 and the following transition probability matrix:

	1	2	3	4
1	0	0	1	0
2	1	0	0	0
3	1/2	1/2	0	0
4	1/3	1/3	1/3	0

i) Find the probability P[X₃, =3,X₂ =1|X₁ = 2].

ii) Classify the states of the given Markov chain.

1 b) It is claimed that the function     is the joint 16

distribution function of the random variables X and Y. Then:

i) determine the corresponding joint probability density function f_x,y and

ii) calculate the probability  

5. a) In an investigation related to a specific type of scores of men and women aged 65 to 70, the mean verbal and performance scores for 101 subjects were found to be:

         

The sample covariance matrix of the scores was

         

In order to test the null hypothesis that observations came from a population with mean vector   apply a suitable test statistic. You may consider α = 0.01 for the test.

b) What is the purpose of principal component analysis? Given the covariance matrix of order 2× ,2 explain how would you extract the principal components. Also, explain how would you find the proportion of total population variance for all the principal components.

6. a) Distinguish between ‘Age Replacement’ and ‘Block Replacement’ policies. Let the lifetimes Y₁ ,Y₂ ,...., are independently and identically distributed random variables and follows negative exponential distribution with parameter 5. If lifetimes T > 0 and age replacement policy is to be employed, then:

i) find the mean renewal time and

ii) find the long-run average cost per unit time, given the costs C 4 1 = and C 6 2 = units of money.

b) In order to fit the regression line y = β₁ + β₂ on a data set consisting of 34 pairs of values (z,y), the least square estimates of β1 and β2 are to be computed. From the data set, the following values are obtained:

Obtain the fitted regression line.

7. a) What is branching process? Give two real examples of branching process.

If P(s) and P_n(s) respectively is the probability generating function (pgf) of the i.i.d. random variables and the random variables {X_n}, where X_n+1 = 

Then show that:

P_n(s) = P_n-1(P(s))

and P_n(s) = P_n-1(P(s)).

b) A community has two police cars, which operate independently of one another. The probability that a specific car will be available when needed is 0.99.

i) What is the probability that neither car is available when needed?

ii) What is the probability that a car is available when needed?

8. State whether the following statements are true or false. Justify your answer with a short proof or a counter example:

i) Although in one-step transition probability matrix, P, of a Markov chain the sum of each row must necessarily be unity but in higher order transition probability matrices, P^(j); j = 2,3,4,...... This rule is not necessary.

ii) For the joint pdf of random variables (X,Y) given by:

iii) Let the r.v. X follows the negative exponential distribution with parameter . Then

iv) One of the examples of non-Poisson queuing system is the (M/G/1): (∞(F1F0) queuing model.

v) If X₁,X₂,..., X, be a random sample from N_p, (μ, Σ), then maximum likelihood estimators of µ and ∑are:

MMT 8 2026 - English

Assignment (MMT – 008)

Course Code: MMT-008
Assignment Code: MMT-008/TMA/2026
Maximum Marks: 100

1. 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 are identical.

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

c) If X₁, X₂, X₃ are iid from , then follows .

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

e) For a renewal function .

2. a) Consider a Markov chain with transition probability matrix:

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.

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

3. a) Let the random vector has mean vector [-2, 3, 4] and variance
covariance matrix . Fit the equation . Also obtain the multiple correlation coefficient between X₃ and [X₁, X₂].

b) Define ultimate extinction in a branching process. Let
0 < b < c < b + c < 1 and . Then discuss the probability of extinction in different cases for or E(X₁) < 1.

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



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 .

iv) Compute  and .

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

ii) Find the marginal distribution of X and Y.

iii) Find the conditional distribution of X given .

iv) Test the independence of variable s X and Y.

v) Find .

5. a) Let , where  and

Find the distribution of:

b) Determine the principal components Y₁, Y₂ and Y₃ for the covariance matrix:

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

6. a) Consider three random variables X₁, X₂, X₃ having the covariance matrix



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.

7. a) If the random vector Z be , where:



and

Find r₃₄, r_34.21.

b) Suppose life times are i.i.d. uniformly distributed on (0, 3) and and . Find:

i)

ii) T which minimizes C(T) and which is the better policy in the long-run in terms of cost.

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

i) Draw the state transition diagram for this chain.

ii) If , then find .

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

iv) Find the stationary distribution for the chain.

b) If N₁(t), N₂(t) are two independent Poisson process with parameters  and  respectively, then show that

, where .

9. a) Let  be a normal random vector with the mean vector  and covariance matrix . Suppose , where

 and .

i) Find .

ii) Compute E(Y).

iii) Find the covariance matrix of Y.

iv) Find .

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  and .

10. 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 (x₁) and demands (x₂) and the following information was obtained:



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



and 

Test whether the sample confirms its truthness of mean vector at  level of significance, if:

i)  is known,
ii)  is unknown.

[You may use: ]

 

 

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