# ISS (Statistical Services) Statistics Paper II (Old Subjective Pattern): Questions 25 - 30 of 39

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## Question 25

Appeared in Year: *2015*

### Describe in Detail

Essay▾Let X = (X_{1} , X_{2} , X_{3}) ‘be distributed as N_{3} (µ, ∑) where µ’ = (2, -3,1) and

(i) Find the distribution of 3X_{1} -2X_{2} + X_{3} .

(ii) Find a 2 × 1 vector a such that X_{2} and are independent.

### Explanation

(i) the distribution of 3X_{1} -2X_{2} + X_{3} is

The mean is

The variance-covariance matrix is

So, the Y has a normal distribution N (13,9) .

(ii) X_{2} and are independent if

## Question 26

Appeared in Year: *2015*

### Describe in Detail

Essay▾To test the hypothesis against for the distribution

Develop the sequential probability ratio test.

### Explanation

Let x_{1} , x_{2} , … , x_{n} are independently and identically distributed with distribution is

Let type I and type II errors are defined as

P (rejecting the lot when θ = θ_{0}) = α

P (accpeting the lot when θ = θ_{1}) = β

Under H_{0} and H_{1} the pmf is

Thus the likelihood ratio is given as

The decision about the acceptance, rejection or continuance of sampling process is…

… (58 more words) …

## Question 27

Appeared in Year: *2015*

### Describe in Detail

Essay▾Let __X__ = (X_{1} , X_{2} , X_{3}) ‘be distributed as N_{3} (__µ__ , ∑) where __µ__ = (10, -7,2) ’ and

Find the partial correlation between X_{1} and X_{2} given X_{3} .

### Explanation

The partial correlation between X_{1} and X_{2} given X_{3} is defined as

where these terms are defined by this equation

Given that

So, the value of equation is

The partial correlation is

## Question 28

Appeared in Year: *2015*

### Describe in Detail

Essay▾Find the maximum likelihood estimator of the 2 × 1 mean vector µ and the 2 × 2 covariance matrix ∑ based on the random sample.

from a bivariate normal distribution.

### Explanation

We known that the maximum likelihood estimator of multivariate normal distribution is

Assume

So,

## Question 29

Appeared in Year: *2015*

### Describe in Detail

Essay▾The observations

3.9,2.4,1.8,3.5,2.4,2.7,2.5,2.1,3.0,3.6,3.6,1.8,2.0,4.0,1.5

are a random sample from a rectangular population with pdf

Estimate the parameters by the method of moments.

### Explanation

Let X_{1} , X_{2} , … , X_{n} be a random sample from a rectangular population. We known that

The estimating equations are

The above equation for solving the parameter, we get

By method of moments, we see the observations

… (1 more words) …

## Question 30

Appeared in Year: *2015*

### Describe in Detail

Essay▾Stating the regularity conditions, give the Cramer-Rao lower bound for the variance of an unbiased estimator of a parameter. Give an example, each, of a situation where the regularity conditions (i) does not hold (ii) holds

### Explanation

Suppose that X_{1} , … , X_{n} is a sample from a distribution with joint pdf f_{n} (__x__ , θ) and T (__X__) is an estimator . Also assume that f_{n} () satisfies the conditions that allow

(i) Interchange of differentiation and integration operations i.e..

(ii) and

(iii)

Then, satisfies the inequality

The example where the regularity condition does not holds

Let X_{1} , … …

… (92 more words) …