# ISS Statistics Paper I (Old Subjective Pattern): Questions 53 - 58 of 165

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## Question number: 53

Appeared in Year: 2015

Essay Question▾

### Describe in Detail

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… (91 more words) …

## Question number: 54

Appeared in Year: 2013

Essay Question▾

### Describe in Detail

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 F (x) =1-S (x) and f (x) =dF (x) /dx

dF (x) =-dS (x)

Hence it’s proofed.

## Question number: 55

» Statistical Methods » Non-Parametric Test » Wilcoxon

Appeared in Year: 2010

Essay Question▾

### Describe in Detail

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… (191 more words) …

## Question number: 56

Appeared in Year: 2012

Essay Question▾

### Describe in Detail

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

## Question number: 57

Appeared in Year: 2011

Essay Question▾

### Describe in Detail

Let X 1, X 2, …, X n be independent N (0, σ 2) random variables. Obtain the mean and variance of . What is its probability distribution?

### Explanation

Let X ~ N (0, σ 2). The density function is

First we find the distribution of Y = X 2

Let

The limit is also change

Differentiate with respect to y,

So, Y = X 2 is follows a… (24 more words) …

## Question number: 58

Appeared in Year: 2013

Essay Question▾

### Describe in Detail

Find the density function of X whose characteristic function Φ (t) is given by:

### Explanation

Rewrite the characteristic function Φ (t)

The density function of X is given by Fourier inversion theorem using the characteristic function,

The general form is

Putting the characteristic function

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