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

Get 1 year subscription: Access detailed explanations (illustrated with images and videos) to **165** questions. Access all new questions we will add tracking exam-pattern and syllabus changes. View Sample Explanation or View Features.

Rs. 550.00 or

## Question number: 27

» Probability » Standard Probability Distributions » Uniform

Appeared in Year: 2010

### 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

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

## Question number: 28

» Probability » Standard Probability Distributions » Cauchy

Appeared in Year: 2012

### Describe in Detail

For the Cauchy distribution given by

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.

The expression of the distribution function is

Assume

The limit is also change.

U = , L= and put

## Question number: 29

» Probability » Sample Space

Appeared in Year: 2011

### 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

## Question number: 30

» Probability » Conditional Probability

Appeared in Year: 2011

### 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

p (0) =p

## Question number: 31

» Probability » Distribution Function » Standard Probability Distributions

Appeared in Year: 2010

### Describe in Detail

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

Find marginal probability density function of X and Y.

### Explanation

The marginal probability density function of X is

The range is 0 < x < 1

The marginal probability density function of Y is

The range is 0 < y < 1

## Question number: 32

» Probability » Tchebycheffs Inequality

Appeared in Year: 2011

### Describe in Detail

Let X be a positive valued random variable. Prove that

Hence deduce the Chebychev’s inequality.

### Explanation

The expectation of X is define as

For any x ≥ r

This implies that

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