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Question:

Let $T: \mathbf{C}^3 \to \mathbf{C}^3 : T \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} x + 2y - iz \\ 2y + iz \\ ix + z - 2z \end{bmatrix}$ . Find [T]_B, [T]_B' and P where
$\quad B = \left\{ \begin{bmatrix} 0 \\ i \\ 0 \end{bmatrix}, \begin{bmatrix} i \\ 1 \\ -1 \end{bmatrix}, \begin{bmatrix} 0 \\ 0 \\ 2 \end{bmatrix} \right\}, B' = \left\{ \begin{bmatrix} 1 \\ -i \\ 1 \end{bmatrix}, \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}, \begin{bmatrix} 1 \\ i \\ 0 \end{bmatrix} \right\}, [T]_{B'} = P^{-1} [T]_B P$ .

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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 $T : V \rightarrow V$ 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 A with characteristic polynomial f(x), $(x - \lambda)^k \mid f(x)$ and $(x - \lambda)^{k+1} \nmid f(x)$ , then the geometric multiplicity of $\lambda$ is at most k.
iv) If $\rho(A) = 1$ , then $A^k \rightarrow \infty$ as $k \rightarrow \infty$ .
v) If N is nilpotent, e^N is also nilpotent.
vi) The sum of two normal matrices of the order n is normal.
vii) If P and Q are positive definite operators, P + Q is a positive definite operator.
viii) Generalised inverse of an $n \times n$ matrix need not be unique.
ix) All the entries of a positive definite matrix are non-negative.
x) The SVD of any $2 \times 3$ matrix is unique.

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Question:

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:
$\sum X=96, \sum Y=72, \sum X^2=780, \sum Y^2=480, \sum XY=588$
The expected mean vector and variance covariance matrix for the factories in the population are:
$\mu = \begin{bmatrix} 9 \\ 7 \end{bmatrix}$
and $\Sigma = \begin{bmatrix} 13 & 9 \\ 9 & 7 \end{bmatrix}.$
Test whether the sample confirms its truthness of mean vector at $5\%$ level of significance, if:
i) $\Sigma$ is known,
ii) $\Sigma$ is unknown.
[You may use: $\chi^2_{2,0.05} = 10.60, \chi^2_{3,0.05} = 12.83, \chi^2_{4,0.05} = 14.89, F_{2,10,0.05} = 4.10$ ]

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Question:

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.

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Question:

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 $P(A \cap B)$ .

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Question:

Let $X = \begin{bmatrix} X_1 \\ X_2 \end{bmatrix}$ be a normal random vector with the mean vector $\mu = \begin{bmatrix} 0 \\ 1 \end{bmatrix}$ and covariance matrix $\begin{bmatrix} 1 & -1 \\ -1 & 2 \end{bmatrix}$ . Suppose $Y = AX + b$ , where
$A = \begin{bmatrix} 1 & 2 \\ 2 & 1 \\ 1 & 1 \end{bmatrix}, b = \begin{bmatrix} 0 \\ 1 \\ 2 \end{bmatrix}$ 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 $P(Y_3 \leq 4)$ .

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Question:

If N₁(t), N₂(t) are two independent Poisson process with parameters $\lambda_1$ and $\lambda_2$ respectively, then show that
$P(N_1(t) = k | N_1(t) + N_2(t) = n) = {}^nC_k p^k q^{n-k}$ , where $p = \frac{\lambda_1}{\lambda_1 + \lambda_2}, q = \frac{\lambda_2}{\lambda_1 + \lambda_2}$

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Question:

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

Image ignouassignments-ignouacademy-com--p-ignou-74776

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.

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Question:

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:
i) $\mu^T$
ii) T which minimizes C(T) and which is the better policy in the long-run in terms of cost.

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Question:

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 $N_4(\mu, \Sigma)$ , where:
$$\mu = \begin{bmatrix} 1 \\ 2 \\ 5 \\ -2 \end{bmatrix}$
and $\Sigma = \begin{bmatrix} 3 & 3 & 0 & 9 \\ 3 & 2 & -1 & 1 \\ 0 & -1 & 6 & -3 \\ 9 & 1 & -3 & 7 \end{bmatrix}.$
Find r₃₄, r_34.21.ii) long-terms rate of failures.

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Question:

ii) long-terms rate of failures.

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Question:

Consider three random variables X₁, X₂, X₃ having the covariance matrix
$$\begin{bmatrix} 1 & 0.12 & 0.08 \\ 0.12 & 1 & 0.06 \\ 0.08 & 0.06 & 1 \end{bmatrix}$
Write the factor model, if number of variables and number of factors are 3 and 1 respectively.

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Question:

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

$$\Sigma = \begin{pmatrix} 1 & -2 & 0 \\ -2 & 5 & 0 \\ 0 & 0 & 1 \end{pmatrix}$

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

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Question:

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 $Y=1$ .

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

Let $X \sim N_3(\mu, \Sigma)$ , where $\mu = [5, 3, 4]'$ and

$$\Sigma = \begin{pmatrix} 2 & 1 & 1 \\ 1 & 1 & 0.5 \\ 1 & 0.5 & 1 \end{pmatrix}$

Find the distribution of:

$$\begin{pmatrix} 2X_1 + X_2 - X_3 \\ X_1 + X_2 + X_3 \end{pmatrix}$

v) Find $V\left[\frac{Y}{X} = x\right]$ .

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Question:

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 $Y=1$ .

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

v) Find $V\left[\frac{Y}{X} = x\right]$ .

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Question:

Let (X, Y) have the joint p.d.f. given by:
$$f(x, y) = \begin{cases} 1, & \text{if } |y| < x; 0 < x < 1 \\ 0, & \text{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 $E(Y | X = x)$ .

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Question:

Let the random vector $X' = (X_1, X_2, X_3)$ has mean vector [-2, 3, 4] and variance
covariance matrix $= \begin{pmatrix} 1 & 1 & 1 \\ 1 & 2 & 3 \\ 1 & 3 & 9 \end{pmatrix}$ . Fit the equation $Y = b_0 + b_1 X + b_2 X_2$ . Also obtain the multiple correlation coefficient between X₃ and [X₁, X₂].
b) Define ultimate extinction in a branching process. Let $p_k = bc^{k-1}, k = 1, 2, \dots;$
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 $E(X_1) \geq 1$ or E(X₁) < 1.

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Question:

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.

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Question:

Consider a Markov chain with transition probability matrix:

Image ignouassignments-ignouacademy-com--p-solve-32906

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.

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Question:

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 X₁, X₂, X₃ are iid from $N_2(\mu, \Sigma)$ , then $\frac{X_1 + X_2 + X_3}{3}$ follows $N_2\left(\mu, \frac{1}{3}\Sigma\right)$ .
d) The partial correlation coefficients and multiple correlation coefficients lie between -1 and 1.
e) For a renewal function $M_t, \lim_{t \to \infty} \frac{M_t}{t} = \frac{1}{\mu}$ .