# ISS (Statistical Services) Statistics Paper I (Old Subjective Pattern): Questions 48 - 54 of 165

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## 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,

… (184 more words) …

## 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) =

Using Simpson’s r

… (157 more words) …

## 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 q = 3/4, then x = 1

… (95 more words) …

## 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

… (292 more words) …

## 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 a random sample, X _{1}, X _{2}, …, X _{n} from ex

… (133 more words) …

## Question number: 53

» Probability » Characteristic Function

Appeared in Year: 2015

### Describe in Detail

Obtain the characteristic function of X whose pdf is

### Explanation

The characteristic function of X is

Let assume x-µ=v, then dx = dv

Assume that , then dv= λ du

First find the value of this integral, this solution adopts the method of contour integration.

Let assume t > 0, For

… (217 more words) …

## Question number: 54

Appeared in Year: 2013

### Describe in Detail

Let X be a continuous random variable and have F (x) as the distribution function. If E [X] exists, then show that:

### Explanation

We known that in continuous random variable, the mean is

We know that F (x) =1-S (x) and f (x) =dF (x) /dx

dF (x) =-dS (x)

Hence it’s proofed.

… (158 more words) …