# 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

Appeared in Year: *2014*

### Describe in Detail

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

### Explanation

Let X_{1} , X_{2} , … , X_{n} , 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, X_{1} , X_{2} , … , X_{n} from exponential distribution with mean parameter θ. Then the d…

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## 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 C_{n} is ni

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

has in the …

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## 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.

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## 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 n_{1} be the number of breakdown times in the Brand A with median time m_{1} , and n_{2} the number of breakdown times in the Brand B with median time m_{2} . 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 …

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## Question 56

Appeared in Year: *2012*

### Describe in Detail

Essay▾Let X_{1} , X_{2} , ···, X_{n} 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