Solve your IGNOU Doubts

Question:

Let E₁, E₂, E₃ and E₄ be arbitrary events. Write the following events in set notations:
$\quad$ i) not more than one of E₁, E₂, E₃, E₄.
$\quad$ ii) one and only one of E₁, E₂, E₃, E₄.
$\quad$ iii) E₁ and at least one of E₂, E₃, E₄.
$\quad$ iv) none of E₂, E₃ and E₄ using E₁.
$\quad$ b) Let the probability density function of r.v. X be
$$f(x) = \begin{cases} 1 + x & ; \ -1 < x \le 0 \\ 1 - x & ; \ 0 < x < 1 \\ 0 & ; \ \text{otherwise} \end{cases}$
$\quad$ and if $u = X$ and $v = X^2$ , find Cov(u, v). Also check the independence of u and v.

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Question:

The mean and standard deviation of a variable x are m and $\sigma$ respectively. Obtain the mean and standard deviation of $\frac{(ax+b)}{c}$ , where a, b and c are constants.
$\quad$ b) If X is a random variable such that $E(X) = 3$ and $E(X^2) = 13$ , determine a lower bound for P(-2 < X < 8).

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Question:

. Consider the joint probability density function

$$f(x, y) = y^2 e^{-y(x+1)}; x \geq 0, y \geq 0.$

Are both x and y regressions linear? Give reasons for your answer.

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Question:

A die is thrown 60 times with the following results:

Face of die	1	2	3	4	5	6
Frequency	8	7	12	8	14	11

Test that the die is unbiased at $5\%$ level of significance. Given that at 5, 6 and 7 d.f. the value of $\chi^2$ are 11.070, 15.592 and 14.067 respectively.

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Question:

6 observations on (X, Y) yielded the following data:

$$\sum X_i = 30, \sum Y_i = 180, \sum X_i Y_i = 1000,$

$$\sum X_i^2 = 200, \sum Y_i^2 = 5642.$

$\quad$ i) Determine the correlation coefficient between X and Y.

ii) Given $X = 10$ , what will be the predicted value of Y?

iii) Given $Y = 15$ , what will be the predicted value of X?

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Question:

The mean I.Q. of a large number of children of age 14 was 100 and standard deviation 16. Assuming that the distribution was normal, find
$\quad$ i) the percentage of children having I.Q. under 80.

ii) the limits in which the I.Q. of the middle 40% of the children will lie.
$\quad$ You may like to use the following values:

$\quad P(Z > 1.25) = 0.1056$

$\quad P(Z < -0.525) = 0.3$

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Question:

Which of the following statements are True or False? Give short proof or counter example in your answer.
i) If the correlation coefficient between X and Y is -0.8, then the correlation coefficient between 2X - 1 and -3Y - 1 is -0.48.
ii) If X and Y are independent binomial variates with parameters (n₁, p₁) and (n₂, p₂) respectively, then X + Y has binomial distribution with parameters (n₁ + n₂, p₁ + p₂).
iii) The function defined as
$$f(x) = \begin{cases} |x| ; & -1 < x < 1 \\ 0 ; & \text{otherwise} \end{cases}$
is a probability density function.
iv) For a normal distribution with mean $\mu$ and variance $\sigma^2$ , the hypotheses
$H_1 : \mu = \mu_0, \sigma^2 = 1$ and
$H_2 : \mu = \mu_0, \sigma^2 \ge 1$ are simple hypotheses.
v) In a problem of testing of a simple hypothesis against a simple alternative, if the probability of type-I error is known to be 0.06, then the power of the test will be 0.94.

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Question:

Evaluate $\displaystyle \iiint_S z^2 \, dx \, dy \, dz$ , where S is the solid region between the spheres $\rho = 1$ and
$\quad$ $\rho = 2$ , by using spherical coordinates.

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Question:

Check if the following integrals are independent of path and evaluate those which are independent.

$$\text{i) } \int_{(0,0)}^{(3,4)} (6xy - y^3) \, dx + (3x^2 - 3xy^2) \, dy$

$$\text{ii) } \int_{(-1,4)}^{(3,8)} (3x^2 - 2y^2) \, dx - 4xy \, dy$

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Question:

Write $\displaystyle \int_0^1 \int_0^{\sqrt{1 - x^2}} (\sqrt{1 - y^2}) \, dy \, dx$ as an integral over a region D. Sketch the region D and show that it is of both types 1 and 2. Reverse the order of integration and evaluate it.

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Question:

Using polar coordinates, show that $\displaystyle \lim_{\substack{x \to 0 \\ y \to 0}} \frac{x^3 - y^3}{x^2 + y^2} = 0$ . Also, find the two repeated limits.

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Question:

Suppose S and C are subsets of $\mathbb{R}^3$ . S is the unit open sphere with centre at the origin and C is the open cube $= \{P(x, y, z) \mid -1 < x < 1, -1 < y < 1, -1 < z < 1\}$ .

Which of the following is true. Justify your answer.
(i) $S \subset C$
(ii) $C \subset S$
(c) Identify the level curves of the following functions:
(i) $\sqrt{x^2 + y^2}$
(ii) $\sqrt{4 - x^2 - y^2}$

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Question:

Find the values of a and b, if

$$\lim_{x \to 0} \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 $\displaystyle \int_C (2x^2 + 3y^2) dx$ , where C is the curve given by

$$x(t) = at^2, y(t) = 2at, 0 \leq t \leq 1.$

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Question:

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