# ISS (Statistical Services) Statistics Paper I (Old Subjective Pattern): Questions 50 - 56 of 164

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## Question 50

Appeared in Year: 2011

### Describe in Detail

Essay▾

Obtain the median and the quartiles of the Cauchy distribution with p. d. f.

### Explanation

For find the median and quartile of the Cauchy distribution, q is any quartile. Then, for which value of q, the x value is

For first quartile q = , then x =-1

For median q = , then x = 0

For third quartile q = , then x = 1

## Question 51

Appeared in Year: 2015

### Describe in Detail

Essay▾

By using Euler-Maclaurin formula, find the sum

### Explanation

The Euler-Maclaurin formula is

where

Given the equation

## Question 52

Edit

Appeared in Year: 2014

### Describe in Detail

Essay▾

Let X1 , X2 , … , Xn be n independently identically distributed random variables following an exponential distribution with mean parameter θ. Obtain the distributions of (i) Maximum (X1 , X2 , … , Xn) ; (ii) Minimum (X1 , X2 , … , Xn) and (iii) Median for n = 2m + 1, (m > 0)

### Explanation

Let X1 , X2 , … , Xn , be n independently identically distributed random variables following an exponential distribution with mean parameter θ. The probability density and distribution function is

Let X(1) , X(2) , … , X(n) denote the order statistic of a random sample, X1 , X2 , … , Xn from exponential distribution with mean parameter θ. Then the d&#8230;

… (81 more words) …

## Question 53

Appeared in Year: 2015

### Describe in Detail

Essay▾

Obtain the characteristic function of X whose pdf is

### Explanation

The characteristic function of X is

Let assume x-µ = v, then dx = dv

Assume that , then dv = λ du

First find the value of this integral, this solution adopts the method of contour integration.

Let assume t > 0, For every n we define in the complex plane the contour Cn is ni

By the residue theorem in the theory of complex we have the function

has in the &#8230;

… (52 more words) …

## Question 54

Appeared in Year: 2013

### Describe in Detail

Essay▾

Let X be a continuous random variable and have F (x) as the distribution function. If E [X] exists, then show that:

### Explanation

We known that in continuous random variable, the mean is

We know that

dF (x) =-dS (x)

Hence it՚s proofed.

… (2 more words) …

## Question 55

Appeared in Year: 2010

### Describe in Detail

Essay▾

The following data represent lifetimes (hours) of batteries for two different brands:

 Brand A 40 30 40 45 55 30 Brand B: 50 50 45 55 60 40

Test whether two brands are the same.

### Explanation

we use Wilcoxon two-sample test to the following data

Let n1 be the number of breakdown times in the Brand A with median time m1 , and n2 the number of breakdown times in the Brand B with median time m2 . Then the hypothesis is that the median times of the two brands are same against the alternative that they are not same.

For this first arrange the &#8230;

… (120 more words) …

## Question 56

Appeared in Year: 2012

### Describe in Detail

Essay▾

Let X1 , X2 , ···, Xn be random variables such that

Also

Find

### Explanation

To find the mean and variance, we use the conditional expectation and conditional variance

By the given definition of conditional expectation

.

.

.

The variance is