Probability (ISS (Statistical Services) Statistics Paper I (Old Subjective Pattern)): Questions 27 - 32 of 72

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Question number: 27

» Probability » Standard Probability Distributions » Uniform

Appeared in Year: 2010

Essay Question▾

Describe in Detail

Let X be a random variable with a continuous distribution function F. Show that F (X) has the uniform distribution on (0, 1).

Explanation

Let U = F (x), then the distribution function of G of U is given by

Equation

The inverse exists. Since F is non-increasing and its continuous

G (u) =F (F -1 (u) ) {F is a distribution of X}

G (u) =u

Then the p. d. f of U

… (37 more words) …

Question number: 28

» Probability » Standard Probability Distributions » Cauchy

Appeared in Year: 2012

Essay Question▾

Describe in Detail

For the Cauchy distribution given by

Equation

where k is a constant to be suitably chosen, derive the expression for the distribution function. Hence obtain a measure of central tendency and a measure of dispersion. What are the points of inflexion of the distribution?

Explanation

We choose k as constant that gives the integral over the range x for the density function is equal to one.

Equation

Equation

Equation

Equation

The expression of the distribution function is

Equation

Equation

Equation

Assume Equation

The limit is also change.

U = Equation , L= Equation and put

… (148 more words) …

Question number: 29

» Probability » Sample Space

Appeared in Year: 2011

Essay Question▾

Describe in Detail

A fair die is thrown until a 6 appears. Specify the sample space. What is the probability that it must be thrown at least 3 times?

Explanation

A fair die content the value {1, 2, 3, 4, 5, 6}

So, the probability of getting 6 is p = 1/6, then probability of getting other than 6 is q = 5/6

If we throw a die, the six is appear. Then the probability is p and the experiment

… (118 more words) …

Question number: 30

» Probability » Conditional Probability

Appeared in Year: 2011

Essay Question▾

Describe in Detail

Consider the following bivariate p. m. f. of (X, Y):

p (0, 10) = p (0, 20) = 2/18;

p (l, 10) = p (l, 30) = 3/18;

p (1, 20) = p (2, 30) = 4/18;

Obtain the conditional mass functions p (y lx = 2), and p (y lx = 1).

Explanation

The joint probability is given

-

x

Total

-

0

1

2

Y

10

2/18

3/18

0

5/18

20

2/18

4/18

0

6/18

30

0

3/18

4/18

7/18

Total

4/18

10/18

4/18

1

First find the marginal probability of x at x = 0, 1, 2

Equation

p (0) =p

… (75 more words) …

Question number: 31

» Probability » Distribution Function » Standard Probability Distributions

Appeared in Year: 2010

Essay Question▾

Describe in Detail

Let (X, Y) be jointly distributed with p. d. f.

Equation

Find marginal probability density function of X and Y.

Explanation

The marginal probability density function of X is

Equation

Equation

Equation

The range is 0 < x < 1

The marginal probability density function of Y is

Equation

Equation

Equation

The range is 0 < y < 1

Question number: 32

» Probability » Tchebycheffs Inequality

Appeared in Year: 2011

Essay Question▾

Describe in Detail

Let X be a positive valued random variable. Prove that

Equation

Hence deduce the Chebychev’s inequality.

Explanation

The expectation of X is define as

Equation

For any x ≥ r

Equation

Equation

Equation

This implies that

Equation

Hence, any function of X, assume g (x) the Chebychev’s inequality is

Equation

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