IGNOU MMT 8 SOLVED ASSIGNMENT

Ques 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 .

Ques 2.

 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.

Ques 3.

 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.

Ques 4.

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.

Ques 5.

 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 .

Ques 6.

 Let the joint probability density function of two discrete random X and Y be given as:

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 .

Ques 7.

 Let the joint probability density function of two discrete random X and Y be given as:

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.

Let , where  and

Find the distribution of:

v) Find .

Ques 8.

 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.

Ques 9.

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.

Ques 10.

 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.

Ques 11.

 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.

If the random vector Z be , where:

and 
Find r₃₄, r_34.21.ii) long-terms rate of failures.

Ques 12.

 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.

Ques 13.

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.

Ques 14.

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

Ques 15.

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 .

Ques 16.

 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 .

Ques 17.

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.

Ques 18.

 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: ]

Ques 19.

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 .

Ques 20.

 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.

Ques 21.

 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.

Ques 22.

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.

Ques 23.

 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 .

Ques 24.

 Let the joint probability density function of two discrete random X and Y be given as:

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 .

Ques 25.

 Let the joint probability density function of two discrete random X and Y be given as:

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.

Let , where  and

Find the distribution of:

v) Find .

Ques 26.

 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.

Ques 27.

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.

Ques 28.

 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.

Ques 29.

 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.

If the random vector Z be , where:

and 
Find r₃₄, r_34.21.ii) long-terms rate of failures.

Ques 30.

 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.

Ques 31.

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.

Ques 32.

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

Ques 33.

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 .

Ques 34.

 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 .

Ques 35.

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.

Ques 36.

 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: ]

Ques 37.

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 .

Ques 38.

 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.

Ques 39.

 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.

Ques 40.

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.

Ques 41.

 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 .

Ques 42.

 Let the joint probability density function of two discrete random X and Y be given as:

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 .

Ques 43.

 Let the joint probability density function of two discrete random X and Y be given as:

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.

Let , where  and

Find the distribution of:

v) Find .

Ques 44.

 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.

Ques 45.

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.

Ques 46.

 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.

Ques 47.

 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.

If the random vector Z be , where:

and 
Find r₃₄, r_34.21.ii) long-terms rate of failures.

Ques 48.

 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.

Ques 49.

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.

Ques 50.

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

Ques 51.

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 .

Ques 52.

 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 .

Ques 53.

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.

Ques 54.

 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: ]

Ques 55.

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 .

Ques 56.

 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.

Ques 57.

 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.

Ques 58.

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.

Ques 59.

 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 .

Ques 60.

 Let the joint probability density function of two discrete random X and Y be given as:

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 .

Ques 61.

 Let the joint probability density function of two discrete random X and Y be given as:

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.

Let , where  and

Find the distribution of:

v) Find .

Ques 62.

 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.

Ques 63.

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.

Ques 64.

 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.

Ques 65.

 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.

If the random vector Z be , where:

and 
Find r₃₄, r_34.21.ii) long-terms rate of failures.

Ques 66.

 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.

Ques 67.

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.

Ques 68.

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

Ques 69.

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 .

Ques 70.

 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 .

Ques 71.

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.

Ques 72.

 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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