Solve your IGNOU Doubts

Question:

A machine component is found to have life-time distribution with probability density function

$f (x) = \frac{x}{100}e^{-x/10} \: \: ,( x> 0) .$

Show that the renewal function is given by

$\frac{t}{20} - \frac{1}{2} + \frac{1}{2} e^{-t/10} ,$

if the process starts with a new component.

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Question:

Suppose the events of a Poisson-process { N (t )} are classified as belonging to category i $( i = 1,2 ......,k)$ with probabilities p_i independently of $N (t) (\sum p_{i} = 1 )$ . Let N (_t) i be the number of events of category i during . $( 0, t ]$ Show that

i) { ${ N , (t)}$ } is a Poisson process with rate $\lambda \: \: p_{i} ,( i = 1 , 2 ...............,k)$

ii) N ₁ are mutually independent.

(Hint: For given $N (t ) = n , ( N_{1} (t),..........., N \: \: \: _{k} (t) )$ are multinomially distribute

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Question:

Using only the definitions, find f_yx (0,0) and f_yx(0,0) if they exists, for the function

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

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Question:

Let ${X}' = [ X_{1} , X_{2}]$ be a random vector with mean vector ${ \mu }'_{x} =[ \mu_{1} ,\mu_{2} ]$ and variance-covariance matrix

$\sum _{x} = \begin{bmatrix} \sigma _{11} &\sigma_{12} \\ \sigma_{12} & \sigma _{22} \end{bmatrix}$

Find the means and covariance matrix for the linear combinations

$Z_{1} = X_{1} - X_{2}$

$Z_{2} = X_{1} + X_{2}$

$Z = \begin{bmatrix} Z_{1} \\Z _{2} \end{bmatrix} =\begin{bmatrix} 1 &-1 \\ 1 &1 \end{bmatrix}\begin{bmatrix} X_{1}\\ X_{2} \end{bmatrix} = CX$

in terms of $\mu _{x}$ and $\sum _{X}$

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Question:

Find the third Taylor polynomial of the function $f(x,y)=1+5xy+3^{2}y\; at\; (1,2).$

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Question:

$(ii)_{x\rightarrow 0+}^{lim}(sin\; x)^{sin\; x}$

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Question:

$(i)_{x\rightarrow +\infty }^{lim}\left ( \frac{x^{2}}{8x^{2}-3} \right )^{1/3}$

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Question:

$(i)\; _{x\rightarrow 0}^{lim}\frac{x^{2}sin\frac{1}{x}}{sin\: x}=1$

(ii) A real-valued function of three variables which is continuous everywhere is differentiable.

(iii) The function F : $\mathbb{R}^{2}\rightarrow \mathbb{R}^{2},$ defined by $F(x,y)=(y+2,x+y)$ , is locally invertible at any $(x,y)\epsilon \; \mathbb{R}^{2}.$

$(iv)f:[-1,1]\times [-2,2]\rightarrow \mathbb{R}, defined\; by$

$f(x,y)=\left\{\begin{matrix} x,\; if\; y\; is\; rational& \\ 0,\; if\; is\; not\; rational & \end{matrix}\right.$

is integrable.

(v) The function $f:\mathbb{R}^{2}\rightarrow \mathbb{R},$ defined by $f(x,y)=1-y^{2}+x^{2},$ has an extremum at (0,0).

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Question:

Check, whether the collection G, given by:

$G'=\left \{] \frac{1}{n+2'} \frac{1}{n}[:n\epsilon \; N\right \}$

is an open cover of ]0,1[.

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Question:

Suppose the joint density function of X and Y is given by

$f ( x, y) = \left\{\begin{matrix} 4y\: (x-y) e^{-(x+y)} \: \: \: \: \: \: \: \: \: 0< x < \infty ,0\leq y \leq x & \\ 0, \; \; \; \; \; \; \; \; \; \; \; \; otherwise& \end{matrix}\right.$

$compute\; E \left [ X \mid Y =y \right ]$

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Question:

Check whether the intervals [5,2] and [7,10 ] are equivalent or not.

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Question:

Show that the sequence (a_n ), an where $a_{n}\frac{n}{n^{2}+4}$ is monotonic. Is (a_{n) sequence? Justify your answer.}

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Question:

State Bonnet’s mean value theorem for integrals. Apply it to show that:

$\left | \int_{3}^{5}\frac{cos\; x}{x}\; dx\right |\leq \frac{2}{3}$

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Question:

i) Calculate the third-degree Taylor polynomial about x = 0 for $f (x) = (1+x)^{1/2}$

ii) Use the polynomial in part (i) to approximate $\sqrt{1.1}$ and find a bound for the error involved.

iii) Use the polynomial in part (i) to approximate $\int_{0}^{0.1} ( 1 + x)^{1/2} dx$

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Question:

Check whether the function f given by:

$f(x)=(x-4)^{3}(x+1)^{2}$

has local maxima and local minima.

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Question:

Using the following table of values, find approximately by Simpson’s rule, the arc length of the graph $y =\frac{1}{x}$ between the points (1, 1) and $\left ( 5,\frac{1}{5} \right )$

x	1	2	3	4	5
$\sqrt{ \frac{1+x^{4} }{x^{4}}}$	1.414	1.031	1.007	1.002	1.001

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Question:

Test the series:

$\sum_{n=1}^{\infty }(-1)^{n-1}\frac{sin\; nx}{n\sqrt{n}}$

for absolute and conditional convergence.

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Question:

Give an example of a series $\sum a_{n}$ such that $\sum a_{n}$ is not convergent but the sequence (a_n) converges to 0.

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Question:

Show that on the curve, y = 3x² − 7x + 6 the chord joining the points whose abscissa are x = 1 and x = 2, is parallel to the tangent at the whose abscissa is $x=\frac{3}{2}$ .