# ISS (Statistical Services) Statistics Paper I (Old Subjective Pattern): Questions 47 - 52 of 165

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## Question number: 47

» Statistical Methods » Correlation Ratio

Appeared in Year: 2012

### Describe in Detail

Scores in a UPSC examination in Statistics Paper-I and Paper-II awarded to10 candidates were as under:

Candidate | A | B | C | D | E | F | G | H | I | J |

Score in Paper- I | 91 | 65 | 80 | 120 | 80 | 140 | 105 | 80 | 42 | 72 |

Score in Paper-II | 86 | 38 | 75 | 135 | 85 | 135 | 94 | 101 | 45 | 65 |

Calculate Spearman’s rank correlation coefficient between the scores in the two papers.

### Explanation

The method of calculating the association between the ranks is known as Spearman’s rank correlation. The candidates are rank by the UPSC examination taken for two papers which is independent. The Spearman’s rank correlation is

where n is the number of candidates and d _{i} is the difference between

## Question number: 48

» 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: 49

» Numerical Analysis » Numerical Integration

Appeared in Year: 2013

### Describe in Detail

Evaluate the following by taking seven ordinates of the integrand:

### Explanation

The seven ordinates of the integrand using Simpson’s rule is

Where h = 1/6 (b-a) and y _{k} =f (a + kh) for k = 0, 1, 2, 3, 4, 5, 6

Here, b = 1, a = 0, h = 1/6, f (x) =

## Question number: 50

» 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: 51

» Numerical Analysis » Summation Formula » Euler-Maclaurin's

Appeared in Year: 2015

### Describe in Detail

By using Euler-Maclaurin formula, find the sum

### Explanation

The Euler-Maclaurin formula is

where

Given the equation

## Question number: 52

» Statistical Methods » Order Statistics » Maximum

Appeared in Year: 2014

### Describe in Detail

Let X _{1}, X _{2}, …, X _{n} be n independently identically distributed random variables following an exponential distribution with mean parameter θ. Obtain the distributions of (i) Maximum (X _{1}, X _{2}, …, X _{n}); (ii) Minimum (X _{1}, X _{2}, …, X _{n}) and (iii) Median for n = 2m + 1, (m > 0)

### Explanation

Let X _{1}, X _{2}, …, X _{n}, be n independently identically distributed random variables following an exponential distribution with mean parameter θ. The probability density and distribution function is

Let X _{ (1) }, X _{ (2) }, …, X _{ (n) } denote the order statistic of