# ISS (Statistical Services) Statistics Paper II (Old Subjective Pattern): Questions 35 - 39 of 39

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## Question number: 35

» Estimation » Optimal Properties » Complete Statistics

Appeared in Year: 2015

### Describe in Detail

Let X _{1}, X _{2}, … X _{n } be a random sample from the binomial distribution with probability mass function

Examine whether the statistic is complete for this distribution.

### Explanation

We know that is sufficient statistic for this distribution. Then,

The definition of completeness is

Assume is free from and

This equation is true iff

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## Question number: 36

» Multivariate Analysis » Multivariate Normal Distribution » Mutliple Correlation Coefficient

Appeared in Year: 2015

### Describe in Detail

Let __X__ _{1}, __X__ _{2}, …, __X__ _{n} be a random sample from an population with , a positive definite matrix. Derive 100 (1-α) % simultaneous confidence interval for for all .

### Explanation

Let __X__ _{1}, __X__ _{2}, …, __X__ _{n} be a random sample from an and assume the linear combination of the random sample is

From, the theorem of linear combinations of multivariate normal distribution is that every linear combination of X follows an univariate normal distribution.

The sa

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## Question number: 37

» Multivariate Analysis » Principal Components » Correlations

Appeared in Year: 2015

### Describe in Detail

Let

Determine

(i) The principal components y _{1}, y _{2} and y _{3}.

(ii) The proportion of variance explained each one of them.

(iii) Correlation between the first principal components y _{1} and the third original random variable.

### Explanation

First find the eigen value and eigen vector pairs

To solve this equation, we get the eigen values is

The corresponding eigen vector for each eigen value is by using normalize the eigen vector by the equation

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## Question number: 38

» Hypotheses Testing » Uniformly Most Powerful Test

Appeared in Year: 2015

### Describe in Detail

Fins a Most powerful test for testing the simple hypothesis against the simple alternative hypothesis based on n random observations from N (µ, σ ^{2}) where µ is know. Show that this MP test is UMP (Uniformly most powerful).

### Explanation

Let X _{i} ‘s are n random observations from N (µ, σ ^{2}) and the pdf is

The hypothesis is

To find the most powerful test, we use following step

The N-P test is

Given α, we can fi

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## Question number: 39

» Multivariate Analysis » Wishart's Distribution » Reproductive Properties

Appeared in Year: 2015

### Describe in Detail

(i) Let A _{i} be distributed as Wishart , i = 1,2 and A _{1}, A _{2} be independent. Show that A _{1} +A _{2} is distributed as .

(ii) If A is distributed as , then CAC’ is distributed as where C is a nonsingular matrix of order m.

### Explanation

(i) We have that is

where ) and Y _{i} ‘s are independent.

Similarly, that is

where ) and Z _{i} ‘s are independent.

We know that A _{1}, A _{2} are independent. So, Y _{i} ‘s are independent of Z _{i} ‘s

Let us defined

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