# ISS (Statistical Services) Statistics Paper I (Old Subjective Pattern): Questions 12 - 17 of 164

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## Question 12

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Appeared in Year: 2014

### Describe in Detail

Essay▾

In a lottery 1000 tickets are sold and the cost of a ticket is if 10. The lottery offers a first prize of if 1,000, two second prizes of if 500 each, and three third prizes of if 100 each. A person purchases a ticket. If X denotes the amount he may get, find E (X) and V (X) .

### Explanation

The probability of purchases a ticket is

First prize amount is 1000

Two second prizes amount is 500 each

Three third prizes amount is 100 each

X denotes the amount he may get

Then, the expected value of X is

The variance of X is

## Question 13

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Appeared in Year: 2010

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Essay▾

The first three moments of a distribution about the value 1 are 2,25 and 80. Find its mean standard deviation and the moment-measure of skewness.

### Explanation

Given that a = 1,

The mean is

The standard deviation is

Measure of skewness

## Question 14

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Appeared in Year: 2013

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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 15

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Appeared in Year: 2010

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Essay▾

Describe clearly sign test. State its asymptotic relative efficiency with respect to t-test.

### Explanation

Let x(1) , x(2) , … , x(n) be the ordered sample values from a population F (X) and M be its median.

Here we test where M0 is the given value of the median and hence

To perform the sign test, first find the differences (X(i) -M0) for i = 1,2, … , n and consider their signs. Suppose the number of + ve sign is r and negative sign is (n-r) . For this w…

… (52 more words) …

## Question 16

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Appeared in Year: 2011

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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…

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## Question 17

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Appeared in Year: 2012

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Essay▾

Two persons Amal and Bimal come to the club at random points of time between 6 p. m. and 7 p. m. , and each stays for 10 minutes. What is the chance that they will meet?

### Explanation

Let x denote the time Amal arrives at the bar and y denote the time Bimal arrives at the bar from 6 p. m. We will draw the lines y = x – 10 and y = x + 10 and shade the area in between these two lines to denote the event. Finding a Ration this Cube

They will meet if that is (x, y) falls in the shaded region. The area of total is 3600 but the total u…

… (40 more words) …