# ISS (Statistical Services) Statistics Paper II (Old Subjective Pattern): Questions 25 - 29 of 39

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## Question number: 25

» Multivariate Analysis » Multivariate Normal Distribution » Mutliple Correlation Coefficient

Appeared in Year: 2015

### Describe in Detail

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 number: 26

» Statistical Quality Control » Sequential Sampling Plans

Appeared in Year: 2015

### Describe in Detail

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}

## Question number: 27

» Multivariate Analysis » Multivariate Normal Distribution » Partial Correlation Coefficient

Appeared in Year: 2015

### Describe in Detail

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 number: 28

» Multivariate Analysis » Estimation » Covariance Matrix

Appeared in Year: 2015

### Describe in Detail

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 number: 29

» Estimation » Estimation Methods » Methods of Moments

Appeared in Year: 2015

### Describe in Detail

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