# 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

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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 …

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## 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…

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## 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.

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

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