# Probability (ISS Statistics Paper I (Old Subjective Pattern)): Questions 20 - 25 of 72

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## Question number: 20

» Probability » Standard Probability Distributions » Geometric

Appeared in Year: 2010

### Describe in Detail

Let X have a geometric distribution, then for an two non-negative integers m and n,

Prove it

### Explanation

This proof is a lack of memory property. We known that

Therefore ………… (1)

The equation is

… (154 more words) …

## Question number: 21

» Probability » Standard Probability Distributions » Cauchy

Appeared in Year: 2011

### Describe in Detail

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

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## Question number: 22

» 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

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## Question number: 23

» Probability » Expectation

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… (135 more words) …

## Question number: 24

» Probability » Expectation

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

… (194 more words) …

## Question number: 25

» 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}

… (243 more words) …