Solve your IGNOU Doubts

Question:

Use least squares method to find a quadratic polynomial that fits the following data:

(-2, 15.7), (-1, 6.7), (0, 2.7), (1, 3.7), (2, 9.7).

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Question:

Let
$$A = \begin{bmatrix} 2 & 2 & 1 \\ -1 & -1 & 2 \\ 0 & 0 & -2 \end{bmatrix}.$
Find a unitary matrix U such that U^*AU is upper triangular.

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Question:

Solve the following system of differential equations:
$\quad \frac{dy(t)}{dt} = Ay(t)$ with $y(0) = \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix}$ , where $A = \begin{bmatrix} 2 & -5 & -11 \\ 0 & -2 & -9 \\ 0 & 1 & 4 \end{bmatrix}$

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Question:

Let M and T be a metro city and a nearby district town, respectively. Our government is trying to develop infrastructure in T so that people shift to T. Each year $15\%$ of T's population moves to M and $10\%$ of M's population moves to T. What is the long term effect on the population of M and T? Are they likely to stabilise?

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Question:

Find the Jordan canonical form J for

$\quad B = \begin{bmatrix} -1 & 0 & -2 & -4 \\ 2 & 1 & 2 & 4 \\ -4 & 2 & -1 & -4 \\ 2 & -1 & 1 & 3 \end{bmatrix}$ .
$\quad$ Also, find a matrix P such that $J = P^{-1}BP$ .

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Question:

If C and D are $n \times n$ matrices such that $CD = -DC$ and D^-1 exists, then show that C is similar to -C. Hence show that the eigenvalues of C must come in plus-minus pairs.

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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-82637

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.