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

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## Question 1

Standard Probability Distributions
Normal
Edit

Appeared in Year: 2013

### Describe in Detail

Essay▾

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.

… (1 more words) …

## Question 2

Standard Probability Distributions
Geometric
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Appeared in Year: 2013

### Describe in Detail

Essay▾

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 said to have lack of memory if

Geometric distribution …

… (80 more words) …

## Question 3

Standard Probability Distributions
Normal
Edit

Appeared in Year: 2011

### Describe in Detail

Essay▾

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 X and Y are independent

which is the moment generating fu…

… (50 more words) …

## Question 4

Standard Probability Distributions
Negative Binomial
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Appeared in Year: 2012

### Describe in Detail

Essay▾

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 r and proportion of item which is defect that is probabil…

… (48 more words) …

## Question 5

Standard Probability Distributions
Geometric

Appeared in Year: 2010

### Describe in Detail

Essay▾

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,

… (1 more words) …

## Question 6

Standard Probability Distributions
Cauchy

Appeared in Year: 2011

### Describe in Detail

Essay▾

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 = , then x =-1

For median q = , then x = 0

For third quartile q = , then x = 1

## Question 7

Standard Probability Distributions
Normal

Appeared in Year: 2011

### Describe in Detail

Essay▾

Let X1 , X2 , … , Xn 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 gamma distribution with parameter .

The additive property of gamma distribution

Then, the mean and variance of gamma distribution is

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