# 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 = (X1 , X2 , X3) ‘be distributed as N3 (µ, ∑) where µ’ = (2, -3,1) and

(i) Find the distribution of 3X1 -2X2 + X3 .

(ii) Find a 2 × 1 vector a such that X2 and are independent.

### Explanation

(i) the distribution of 3X1 -2X2 + X3 is

The mean is

The variance-covariance matrix is

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

(ii) X2 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 x1 , x2 , … , xn are independently and identically distributed with distribution is

Let type I and type II errors are defined as

P (rejecting the lot when θ = θ0) = α

P (accpeting the lot when θ = θ1) = β

Under H0 and H1 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 = (X1 , X2 , X3) ‘be distributed as N3 (µ , ∑) where µ = (10, -7,2) ’ and

Find the partial correlation between X1 and X2 given X3 .

### Explanation

The partial correlation between X1 and X2 given X3 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 X1 , X2 , … , Xn 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

Cramer-Raoinequality
Edit

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 X1 , … , Xn is a sample from a distribution with joint pdf fn (x , θ) and T (X) is an estimator . Also assume that fn () 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 X1 , … …

… (92 more words) …