# ISS (Statistical Services) Statistics Paper I (Old Subjective Pattern): Questions 11 - 16 of 165

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## Question number: 11

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Appeared in Year: 2012

Essay Question▾

### Describe in Detail

Show, by proving all intermediate results, that for a set of n unsorted data, the measure of skewness based on mean, median and standard deviation, necessarily lies between -3 and + 3.

### Explanation

The formula of measure of skewness given by Karl Pearson’s is

Also, we known the relation of mean, median and mode is

Mean-Mode = 3 (Mean-Median)

So,

Using Chebyshev’s inequality, X be a random variable with mean µ and variance

… (163 more words) …

## Question number: 12

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Appeared in Year: 2014

Essay Question▾

### Describe in Detail

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

… (91 more words) …

## Question number: 13

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Appeared in Year: 2010

Essay Question▾

### Describe in Detail

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

… (52 more words) …

## Question number: 14

» Probability » Standard Probability Distributions » Geometric

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Appeared in Year: 2013

Essay Question▾

### 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 said to have lack of memory if

… (299 more words) …

## Question number: 15

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Appeared in Year: 2010

Essay Question▾

### Describe in Detail

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 M 0 is the given value of the median and hence

To perform the sign test, first find the differences (X (i) -M 0) for i = 1,2, …, n and consider their signs. Su

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## Question number: 16

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Appeared in Year: 2011

Essay Question▾

### 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

… (103 more words) …

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