Solve your IGNOU Doubts

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.

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

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.

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Question:

State giving reasons, if the following statement are true or false.

a) A closed map on a normed space need not be an open map.

b) c₀₀ is a closed subspace of $l^\infty$

c) The dual of a finite dimensional space is finite dimensional.

d) If T₁ and T₂ are positive operators on a Hilbert space H, then T₁ + T₂ is a positive operator on H.

e) On a normed space X, the norm function $||.|| : X \to C$ is a linear map.

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Question:

Let X be a Banach space and Y be a closed subspace of .X Let $\pi:X \to X/Y$ be canonical quotient map. Show that $\pi$ is open.

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Question:

Define the spectral radius of a bounded linear operator $A \in BL(X)$ .Find the spectral radius of A in $BL(R^3)$ ,where A is given by the matrix

$\begin{bmatrix} 0 & 1 & 0 \\ -1 & 0 & 0\\ 0& 0 & -1 \end{bmatrix}$

with respect to the standard basis of $R^3$ .

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Question:

Let {A_n} be a sequence of unitary operators in BL(H). Prove that if A $||A_n - A|| \to 0, A \in BL(H)$ then A is unitary.

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Question:

Define a Hilbert-Schmidt operator on a Hilbert space H and give an example. Is every Hilbert-sehmidt operator a compact operator? Justify your answer.

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Question:

Let X,Y be normed spaces and suppose BL(X,Y) and CL(X,Y) denote, respectively, the space of bounded linear operators from X to Y and the space of compact linear operators from X to Y. Show that CL(X,Y) is linear subspace of BL(X,Y). Also, Show that if Y is a Banach space, then CL(X,Y) is a closed subspace of BL(X,Y).

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Question:

Let X be a normed space and Y be proper subspace of X. Show that the interior Y⁰ of Y is empty

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Question:

Give one example of each of the following. Also justify your choice of example.

i) A self-adjoint operator on $l^2$ .

ii) A normal operator on a Hilbert space which is not unitary.

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Question:

Let $X = c_{00}$ with $||.||_p$ Give an example of a Cauchy sequence in X that do not converge in X. Justify your choice of example

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Question:

Let A be a normal operator on a Hilbert space X. Show that $\sigma(A) \subset \sigma_a(A)$ where $\sigma_a(A)$ denotes the approximate eigen spectrum of A and $\sigma(A)$ denotes the spectrum of A.

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Question:

Given an example of an Hilbert space H and an operator A on H such that $\sigma_e(A)$ is empty. Justify your choice of example.

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Question:

Let $H = R^3$ and F be the set of all $x = (x_1,x_2,x_3)$ in $H$ such that $x_1 = 0$ . Find $F^\perp$ Verify that every $x \in H$ can be expressed as $x = y + z$ where $y \in F$ and $z \in F^\perp$ .

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Question:

Let X be an inner product space and $x,y \in X$ Prove that $x \perp y$ if and only if $||kx + y||^2 = ||kx||^2 + ||y^2||, k \in K$

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Question:

Let $f : C[0,1] \to \mathbb{R}$ be given by $f(x) =x(1)\forall \: x \in C[0,1]$ Show that f is continuous w.r.t the supnorm and f is not continuous w.r.t the p-norm.

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Question:

Which of the following maps are open? Give reasons for your answer.

i) $T : \mathbb{R}^3 \to \mathbb{R}^2$ given by T(x,y,z) = (x,z)

ii) $T : \mathbb{R}^3 \to \mathbb{R}^3$ given by T(x,y,z) = (x,y,0)

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Question:

Read the proof of the closed graph theorem carefully and explain where and how we have used the following facts in the proof.
i) X is a Banach space.
ii) Y is a Banach space.
iii) F is a closed map.

iv) Which property of continuity is being established to conclude that F is continuous.

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