# Probability-Characteristic Function (ISS (Statistical Services) Statistics Paper I (Old Subjective Pattern)): Questions 1 - 3 of 3

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## Question number: 1

» 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: 2

» 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

## Question number: 3

» Probability » Characteristic Function

Appeared in Year: 2009

### Describe in Detail

Find the density, if its characteristic function is

### 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