Solve your IGNOU Doubts

Question:

Which of the following statements are true and which are false? Give reasons for your answer

i) If $V$ is a finite dimensional vector space and $T:V \to V$ is a diagonalisable linear operator, then there is a basis, unique up to order of the elements, with respect to which the matrix of $T$
is diagonal.
ii) Up to similarity, there is a unique $3 \times 3$ matrix with minimal polynomial $(x-1)^2(x-2)$ .
iii) If $\lambda$ is the eigenvalue of a matrix See Answer →

Question:

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 $\Sigma$

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Question:

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

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Question:

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

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Question:

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

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

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Question:

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.

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Question:

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 $p_2 = \frac{1}{6}$

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Question:

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

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Question:

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.

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Question:

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, $\chi_{9,0.05}^{2} = 3.84$ , $\chi_{10,0.05}^{2} = 4.10$ , $\chi_{11,0.05}^{2} = 5.09$ ]

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Question:

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, $M_t$ .

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Question:

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

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Question:

Write two advantages and two disadvantages of conjoint analysis.

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Question:

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.

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Question:

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).

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Question:

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.

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Question:

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

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Question:

Consider the Markov chain having the following transition probability matrix.

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