Solve your IGNOU Doubts

Question:

Consider the two data sets

$X_{1} = \begin{bmatrix} 1 &2 &4 \\ 5 &3 &5 \\ \end{bmatrix} and \: X_{2} =\begin{bmatrix} 3 & 2 & 4\\ 9 & 5& 7\\ \end{bmatrix}$

for which

$_{x_{1}}^{-}= \begin{bmatrix} 2\\ 4 \end{bmatrix} ,=_{x_{2}}^{-} =\begin{bmatrix} 3\\ 5 \end{bmatrix}$

and

$S \: \: \: _{pooled} =\begin{bmatrix} 1 & 1\\ 1& 2 \end{bmatrix}$

i) Calculate the linear discriminant function.

ii) Classify the observation $X_{0}^{{}'} = [ 2 , 7]$ as population $\pi_{1}$ or population $\pi_{2}$ with equal priors and equal costs.

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Question:

Find the values of a and , b if

$_{x\rightarrow \infty }^{lim}\frac{x(1+a\; cos\; x)-b\; sin\; x}{x^{3}}=1$

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Question:

Use double integration of find the volume of the ellipsoid

$\frac{x^{2}}{4}+\frac{y^{2}}{9}+\frac{z^{2}}{16}=1.$

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Question:

Evaluate $\int_{C}^{}(2x^{2}+3y^{2})$ dx, where C is the curve given by $x(t)=at^{2},y(t)=2at,0\leq t\leq 1.$

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Question:

Twenty-five portfolio managers were evaluated in terms of their performance. Suppose Y represents the rate of return achieved over a period of time; Z₁ is the manager’s attitude toward risk measured on a five-point scale from “very conservative” to “very risky”; and Z₂ is years of experience in the investment business. The observed correlation coefficients between pairs of variables are

$R = \begin{bmatrix} Y & Z_1 &.Z_2 \\ 1.0 & -.35 &.82 \\ -.35 &1.0 & -.60\\ .82&-.60 & 1.0 \end{bmatrix}$

i) Interpret the sample correlation coefficients $r_{yz_{1}} = -.35$ and $r_{yz_{2}} = -. 82$

ii) Calculate the partial correlation coefficient $r_{yz_{1}} z_{2}$ and interpret this quantity with respect to the interpretation provided for $r_{yz_{1}}$ in Part (i)

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Question:

Find the domain and range of the function f , defined by $f(x,y)=\frac{2xy}{x^{2}+y^{2}}.$ Also find two level curves of this function. Give a rough sketch of them.

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Question:

Check the continuity and differentiability of the function at (0,0) where

$f(x,y)=\left\{\begin{matrix} \frac{2x^{3}y}{x^{2}+y^{2}},\; \; (x,y) \neq (0,0)& \\ 0,\; \; \; \; \; \; \; \; \; otherwise & \end{matrix}\right.$

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Question:

Consider a birth and death process with birth rate $\lambda\: \: _{i} =( i+1)\lambda , i\geq 0$ and death rates $\mu \: \: _{i} = i\mu , i\geq 0$ .

i) Determine the expected time to go from state 0 to state 4.

ii) Determine the expected time to go from state 2 to state 5.

iii) Determine the variances in parts (i) and (ii)

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Question:

Check the local inevitability of the function f defined by f(x,y) = (x² - y², 2xy) at (1,-1) Find a domain for the function f in which f is invertible.

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Question:

State a necessary condition for the functional dependence of two differentiable functions f and g on an open subset D of $\mathbb{R}^{2}$ Verify this theorem for the functions f and g, defined by

$f(x,y)=x^{5}+y^{5}-16xy^{3}-1=0$

defines the function F. Also find g′( y).

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Question:

Suppose that people arrive at a bus stop in accordance with a Poisson process with rate $\lambda$ . The bus departs at time t . Let_^Xdenote the total amount of waiting time of all those that get on the bust at time t . We want to determine Var ( X ) . Let N (t ) denote the number of arrivals by time t

i) What is E [ X | N (t)] ?

ii) Argue that Var $[ x \mid N (t) ] = N (t) t^{2} /12$

iii) What is Var ( X ) ?

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Question:

State Green’s theorem, and apply it to evaluate $\int_{c}^{}(3x^{2}-4y)dx-(2x+y^{3})dy$

where C is the ellipse 4x²+ 9y² = 36.

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Question:

For a branching process, calculate $\pi _{o}$ when

$i ) \: \: P_{o} = \frac{1}{5},\: \: P_{2} = \frac{4}{5}$

$i i) \: \: P_{o} = \frac{1}{5},\: \: P_{1} = \frac{3}{5} P_{2}= \frac{1}{5}$

$ii i) \: P_{o} = \frac{1}{6},\: \: P_{1} = \frac{1}{2} P_{3}= \frac{1}{3}$

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Question:

Find the mass of the solid bounded by $z$ = 1 and , $z$ = x² + y² the density function being δ (z,y,x) =.|x|

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Question:

Find the centre of gravity of a thin sheet with density δ(x, y) = y, bounded by the curves y = 4x² and x = 4.

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Question:

Let a - (1,2,3),b - (-5, 3, -2),c-(2,-4,1) be three points in $\mathbb{R}^{3}.$ Find | 2b − a + 3c |.

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Question:

For a series of dependent trials the probability of success on any trial is $(k+1) / ( k+2)$ where_k is equal to the number of successes on the previous two trials. Compute lim $_{n\rightarrow \infty }P$ [success on the n th trial].

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Question:

Let x = e' cos θ, y = e' sin θ and f be a continuously differentiable function of x and y, whose partial derivatives are also continuously differentiable. Show that

$\frac{\partial ^{2}f}{\partial r^{2}}+\frac{\partial ^{2}f}{\partial \theta ^{2}}=(x^{2}+y^{2})\left ( \frac{\partial ^{2}f}{\partial x^{2}}+\frac{\partial ^{2}f}{\partial y^{2}} \right )$