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IGNOU MSCMACS MMT 8 2024 Solution

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Last Date of Submission of IGNOU MMT-08 (MSCMACS) 2024 Assignment is for January 2024 Session: 30th September, 2024 (for December 2024 Term End Exam).
Semester Wise
January 2024 Session: 30th March, 2024 (for June 2024 Term End Exam).
July 2024 Session: 30th September, 2024 (for December 2024 Term End Exam).

Title Name	ignou MMT 8 solved assignment 2024
Type	Soft Copy (E-Assignment) .pdf
University	IGNOU
Degree	MASTER DEGREE PROGRAMMES
Course Code	MSCMACS
Course Name	M.Sc. Mathematics with Applications in Computer Science
Subject Code	MMT 8
Subject Name	Probability and Statistics
Year	2024
Session	-
Language	English Medium
Assignment Code	MMT-08/Assignmentt-1//2024
Product Description	Assignment of MSCMACS (M.Sc. Mathematics with Applications in Computer Science) 2024. Latest MMT 08 2024 Solved Assignment Solutions
Last Date of IGNOU Assignment Submission	Last Date of Submission of IGNOU MMT-08 (MSCMACS) 2024 Assignment is for January 2024 Session: 30th September, 2024 (for December 2024 Term End Exam). Semester Wise January 2024 Session: 30th March, 2024 (for June 2024 Term End Exam). July 2024 Session: 30th September, 2024 (for December 2024 Term End Exam).
Assignment Code	MMT 8/2024

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Questions Included in this Help Book

Ques 1.

Consider the Markov chain having the following transition probability matrix.

$P = \begin{bmatrix} \frac{1}{3} & \frac{2}{3} & 0 & 0 & 0 & 0 \\ \frac{2}{3} & \frac{1}{3} & 0 &0 &0 &0 \\ \frac{1}{4} & 0 & \frac{1}{4} & 0 & \frac{1}{4} & \frac{1}{4} \\ \frac{1}{6} & \frac{1}{6} & \frac{1}{6} & \frac{1}{6} & \frac{1}{6} & \frac{1}{6} \\ 0& 0 & \frac{1}{4} & \frac{3}{4} & 0& 0\\ 0& 0 & \frac{1}{5} & \frac{4}{5} & 0 & 0 \end{bmatrix}$

i) Draw the diagram of a Markov chain.

ii) Classify the states of a Markov chain, i.e., persistent, transient, non-null and a periodic state. Also check the irreducibility of Markov chain.

iii) Find the closed sets.

iv) Find the probability of absorption to the closed classes. Also find the mean time up to absorption from transient state 3 to 4.

Ques 2.

Determine the parameters of the bivariate normal distribution:

$f(x,y) = k exp \left [ -\frac{8}{27} \left \{ (x-7)^2 - 2(x-7)(y+5) + 4(y+5)^2 \right \}\right ]$

Also find the value of k

Ques 3.

Suppose that the probability of a dry day (State 0) following a rainy day (State 1) is 1/3 and the probability of a rainy day following a dry day is 1/2. Write the transition probability matrix of the above Markov chain.

Given that 1^st May is a dry day, then calculate

i) the probability that 3^rd May is also a dry day.

ii) the stationary probabilities.

Ques 4.

Let $X \sim N_4 (\mu ,\Sigma )$ with

$\mu = \begin{pmatrix} 2 \\ 1 \\ 3 \\ -4 \end{pmatrix}$ and

$\Sigma \begin{bmatrix} 1 & 1 & 1 & 1\\ 1& 2& -2 & -1\\ 1& -2 & 9 & -1 \\ 1& -1 & -1 & 16 \end{bmatrix}$

Support Y and Z are two partitioned subvectors of X such that Y' = (x₁x₃) and Z' = (x₂x₄)

i) Obtain the marginal distribution of Y′.

ii) Check the independence of Y′ and Z′.

iii) Obtain the conditional distribution of Y′ | Z′ ; where Y' = (x₁x₂) , Z' = (x₃x₄)

iv) Find E(Y' | Z' )]; where Y′ and Z′ are same as in (iii).

Ques 5.

Suppose that customers arrive at a service counter in accordance with a Poisson process with the mean rate 2 per minute. Then obtain the probability that the interval between two successive arrivals is
i) more than 1 minute.
ii) 4 minutes or less.
iii) between 1 and 2 minutes.

Ques 6.

Write two advantages and two disadvantages of conjoint analysis.

Ques 7.

Find the differential equation of pure birth process with $\lambda_K = K\lambda$ and the process start with one individual at time t = .0 Hence, find $p_n(t) = P(N(t) = n)$ [N(t) is the number present at time t] with $E(N(t))$ and $Var(N(t))$ . Also identify the distribution

Ques 8.

Let $\left \{ X_n; n\geq1 \right \}$ be an i.i.d. sequence of interoccurrence times with common probability mass function given by
$P(X_n = 0) = \frac{2}{3}, P(X_n = 1) = P(X_n = 2) = \frac{1}{6}$
Let $N_t; t \geq 0$ be the corresponding renewal process. Find the Laplace transform $M_t$ of the renewal function, $M_t$ .

Ques 9.

The body dimensions of a certain species have been recorded. The information of body length L and body weight W are given below:

Body length L (in mm)	Body weight W (in mg)
45	2.9
48	2.4
45	2.8
48	2.9
44	2.4
45	2.3
45	3.1
42	1.7
50	2.4
52	3.7

At 5% level of significance, test the hypothesis that all variances are equal and all covariances are equal in variance-covariance matrix for the given data.

[You may like to use the values, $\chi_{9,0.05}^{2} = 3.84$ , $\chi_{10,0.05}^{2} = 4.10$ , $\chi_{11,0.05}^{2} = 5.09$ ]

Ques 10.

The Tooth Care Hospital provides free dental service to the patients on every Saturday morning. There are 3 dentists on duty, who are equally qualified and experienced. It takes on an average 20 minutes for a patient to get treatment and the actual time taken is known to vary approximately exponentially around this average. The patients arrive according to the Poisson distribution with an average of 6 per hour. The officer of the hospital wants to investigate the following:
i) The expected number of patients in the queue.
ii) The average time that a patient spends at the clinic.
iii) The average percentage idle time for each of the dentists.

Ques 11.

For the two-state Markov chain, whose transition probability matrix is

$P = \begin{pmatrix} 1-p & p \\ p & 1-p \end{pmatrix} ; 0\leq p \leq 1$

Find all stationary distributions

Ques 12.

Let p_K ,where K = 0,1,2 be the probability that an individual generates K offsprings. Then find the p.g.f. of {p_K } . Also, calculate the probability of extinction when

i). $p_0 = \frac{1}{4}, p_1 = \frac{1}{4}$ and $p_2 = \frac{1}{2}$

ii). $p_0 = \frac{2}{3}, p_1 = \frac{1}{6}$ and $p_2 = \frac{1}{6}$

Ques 13.

Let p = 3 and m =1 and suppose the random variables X₁, X₂ and X₃ have the positive definite covariance matrix:

$\Sigma = \begin{bmatrix} 1 & 0.4 & 0.3 \\ 0.4 & 1 & 0.2 \\ 0.3 & 0.2 & 1 \end{bmatrix}$

Write its factor model.

Ques 14.

For X distributed as $N_3(\mu,\Sigma )$ find the distribution of

$\begin{bmatrix} X_1 & -X_2 & X_3 \\ -X_1 & X_2 & X_3 \end{bmatrix}$

Ques 15.

The joint density function of random variables X, Y and Z is given as $f(x,y,z) = K.x.e^{-(y+z)}$ ;
where $0< x< 2, y\geq 0$ and $z\geq 0$

Find
i) the constant K.
ii) the marginal distributions of X, Y and Z
iii) E(X), E(Y) and E(Z)
iv) the conditional expectation of Y given X and Z.
v) the correlation coefficient between X and Y

Ques 16.

For the model M | M |1| N | FIFO, calculate the steady state solution for P₀ .

E(n) − Average number of customers in the system

E(V) – Average waiting time in the system

Ques 17.

State which of the following statements are true and which are false. Give a short proof or a counter example in support of your answer.

a) For three independent events E₁ , E₂and E₃,

$P(E_1\cup E_2 \cup E_3) + P(\bar{E_1})P(\bar{E_2})P(\bar{E_3}) = 0$

b) The range of multiple and partial correlation coefficient is $]-1,1[$
c) If $\left \{ X(t); t \geq 0 \right \}$ is a poisson process, then $N(t) = [X(t + S_0) - X(t)]$ where $S_0 > 0$ is a
fixed constant, is also a poisson process.
d) In Hotelling T² , the value of S is given by

$S = \frac{1}{n-1}\sum_{j=1}^{n}(X_j - \mu)(X_j - \mu)'$
e) Let $X_{p\times 1} \sim N_p(\mu,\Sigma)$ and $X_{p\times n}$
be the state matrix, then parameters involved in the above
distribution are p for $\mu$ and $\frac{1}{2}p(p+1)$ for $\Sigma$