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

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## Question number: 53

» Probability » Characteristic Function

Appeared in Year: 2015

### 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

## Question number: 54

Appeared in Year: 2013

### 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

### 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

## Question number: 56

Appeared in Year: 2012

### 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

» Probability » Standard Probability Distributions » Normal

Appeared in Year: 2011

### 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

## Question number: 58

» Probability » Characteristic Function

Appeared in Year: 2013

### 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