# Probability-Probability Generating Functions (ISS (Statistical Services) Statistics Paper I (Old Subjective Pattern)): Questions 1 - 4 of 4

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## Question number: 1

» Probability » Probability Generating Functions

Appeared in Year: 2013

### Describe in Detail

Let P (z) be the probability generating function of the random variable X whose probability distribution is Pr [X = n] = p _{n}, n = 0, I, 2, …… Find the generating function of (i) Pr [X > n],

(ii) Pr [X < n] and (iii) Pr [X = 2n].

### Explanation

Let P (z) be the probability generating function of the random variable X whose probability distribution is Pr [X = n] = p _{n}. Then

(i) The generating function of Pr [X > n] is denoted by P (z _{ > n})

The generating function is

## Question number: 2

» Probability » Probability Generating Functions

Appeared in Year: 2009

### Describe in Detail

Given the distribution function

find its probability density function.

### Explanation

The distribution function written in this form

The density function is

where

## Question number: 3

» Probability » Probability Generating Functions

Appeared in Year: 2014

### Describe in Detail

Let P (s) be the probability generating function associated with a non-negative, integer valued random variable X. Show that

### Explanation

To show this we use the left hand side, express the summation term

We know that the probability generating function is defined as

Putting this in the above equation, we prove the following

## Question number: 4

» Probability » Probability Generating Functions

Appeared in Year: 2009

### Describe in Detail

Find the generating function of X whose probability density function is

P [X = r] =pq ^{r-1}, r = 1, 2, …, 0 < p < 1, q = 1 - p

### Explanation

The probability generating function is defined as