# ISS (Statistical Services) Statistics Paper I (Old Subjective Pattern): Questions 35 - 40 of 165

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## Question number: 35

» Probability » Elements of Measure Theory

Appeared in Year: 2013

### Describe in Detail

If the probability for n independent events are p _{1}, p _{2}, …, p _{n}, then prove that:

(i) none of the events will occur

(ii) at least one event will occur

(iii) at most one event will occur;

### Explanation

Let A _{1 }, A _{2}, …, A _{n}, n independent events with probabilities p _{1}, p _{2}, …, p _{n}.

(i) Find the probability that none of the events will occur

Since events are independent and the probability that one event does not occur

## Question number: 36

» Statistical Methods » Non-Parametric Test » Run

Appeared in Year: 2012

### Describe in Detail

Name two non-parametric tests for comparing locations of two correlated populations.

### Explanation

The Wald-Wolfowitz runs test and Mann-Whitney U-test comparing locations of two correlated populations. Let the sample 1 consider the population X and sample 2 consider the population Y. The hypothesis whether two sample comes from same identical population. We test the following null and alternative hypothesis is

## Question number: 37

» Statistical Methods » Non-Parametric Test » Wald-Wolfowitz

Appeared in Year: 2013

### Describe in Detail

The values of two random samples are arranged below in an increasing order with X denoting the value of sample I and Y denoting the value of sample 2:

XXXYXYYYYXXYXXXYXXYYY

Use Wald-Wolfowitz run test to test whether the two samples may be regarded as coming from a common population.

### Explanation

Wald-Wolfowitz run test uses the data of two random samples, sample 1 is X variables of size n = 11 and sample 2 is Y variables of size m = 10 from two population F _{X} (x) and F _{Y} (y) respectively.

The hypothesis under the test is that the

## Question number: 38

» Probability » Probability Generating Functions

Appeared in Year: 2009

### Describe in Detail

Given the distribution function

find its probability density function.

### Explanation

The distribution function written in this form

The density function is

where

## Question number: 39

» Probability » Convergence » In Distribution

Appeared in Year: 2013

### Describe in Detail

Show that convergence in probability implies convergence in distribution.

### Explanation

The sequence X _{n} converges to X in probability if for any ε > 0

The sequence X _{n} converges to the distribution of X as n tends to infinity if

For ε > 0,

Hence

……… (1)

Note that

…. (2)

From

## Question number: 40

» Probability » Probability of M Events Out of N

Appeared in Year: 2009

### Describe in Detail

A die is rolled twice. Let A, B, C denote the events respectively that the sum of scores is 6, the sum of scores is 7, and the first score is 4. Are A and C independent? Are B and C independent?

### Explanation

A die is rolled twice, the sample space consists of thirty six outcomes. The sample space is

(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6)

(2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6)

(3, 1), (3, 2), (3, 3), (3, 4), (3,