# Probability-Standard Probability Distributions (ISS (Statistical Services) Statistics Paper I (Old Subjective Pattern)): Questions 1 - 7 of 22

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## Question number: 1

» Probability » Standard Probability Distributions » Normal

Appeared in Year: 2013

### Describe in Detail

If X ~ N (0, 1), obtain the distribution of X ^{2}.

### Explanation

X ~ N (0, 1). The density function is

Let assume Y = X ^{2}

Let

The limit is also change

Differentiate with respect to y,

So, Y = X ^{2} is follows a chi-square distribution with one degree of freedom.

## Question number: 2

» Probability » Standard Probability Distributions » Geometric

Appeared in Year: 2013

### Describe in Detail

Prove that for among the discrete distributions, the geometric distribution has the lack of memory property.

### Explanation

The property of memory less is that these distributions of “time from now to the next period” are exactly the same. The property is most easily explained in terms of “waiting times.

Suppose X is a discrete random variable whose values is a non-negative. In probability theory, a distribution is

## Question number: 3

» Probability » Standard Probability Distributions » Normal

Appeared in Year: 2011

### Describe in Detail

Prove that the sum of two independent chi-squared random variables is also chi-squared.

### Explanation

Let X and Y are two independent chi-squared random variables with degree of freedom n and m respectively. We proof this by moment generating function. The moment generating function of chi-squared distribution is

Then moment generating function of sum of two random variable (X + Y) is

because

## Question number: 4

» Probability » Standard Probability Distributions » Negative Binomial

Appeared in Year: 2012

### Describe in Detail

Items from a large lot are examined one by one until r items with a rare manufacturing defect are found. The proportion of items with this type of defect in the lot is known to be p. Let X denote the number of items needed to be examined. Derive the probability distribution of X, and find E (X).

### Explanation

In this question, the sample size is n = x+r given and each trail only two possible outcomes. The probability of defect is same for each trail and trails are independent. The experiment continues until r defectives.

In the given question the number of manufacturing defect are fixed which is

## Question number: 5

» 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

Using equation (1)

Since the equation is true. Therefore,

## Question number: 6

» 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

For first quartile q = 1/4, then x=-1

For median q = 1/2, then x = 0

For third quartile

## Question number: 7

» 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