# ISS (Statistical Services) Statistics Paper II (Old Subjective Pattern): Questions 20 - 25 of 39

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## Question number: 20

» Estimation » Estimation Methods » Maximum Likelihood

Appeared in Year: 2014

### Describe in Detail

Let X be exponentially distributed with parameter θ. Obtain MLE of θ based on a sample of size n, from the above distribution.

### Explanation

Let X be exponentially distributed with parameter θ.

The likelihood function is

The log-likelihood function is

Differentiable with respect to θ, equating to zero

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## Question number: 21

» Estimation » Optimal Properties » Confidence Interval Estimation

Appeared in Year: 2015

### Describe in Detail

Let y _{1}, y _{2, …, } y _{n} be a random sample from N (µ, σ ^{2}) where µ and σ ^{2} are both unknown. Obtain a confidence interval of µ with confidence coefficient (1-α)

### Explanation

When population mean and population standard deviation in not know. If is the samplemean and replace σ by its estimate s and t _{α/2} be the critical value of the student t-test such that have of the area on the left hand side and other half on the right side that is

For first inequality we get

From secon

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## Question number: 22

» Estimation » Optimal Properties » Sufficient Estimator

Appeared in Year: 2015

### Describe in Detail

Obtain the sufficient statistics for the following distribution.

(i)

(ii)

### Explanation

By using factorization theorem, the condition is that

where h (x) is free from θ and g (. ) depends on __X__ only through T.

(i)

The joint pdf of random sample is

Let T= . By factorization theorem

h (x) =1,

So, h (x) is free from θ

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## Question number: 23

» Multivariate Analysis » Distribution of Hotelling T2 Statistic » Use for Testing

Appeared in Year: 2015

### Describe in Detail

Show that T ^{2} statistic is invariant under changes in the unit of measurements for a p×1 random vector X of the form __Y__ =C __X __ + __d __ where C is a p×p nonsingular matrix, __d__ is a p×1 vector.

### Explanation

T ^{2} statistic can be computed from __X__. Here we have to compute it form __Y__ and both will be same. Here

Let , then and

So, T ^{2} -statistic comes into existence for testing hypothesis of the form

Since is known vector,

If we have a setup

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## Question number: 24

» Statistical Quality Control » Concepts of ATI

Appeared in Year: 2015

### Describe in Detail

For a sequential probability ratio test of strength (α, β) and stopping bounds are A and B (B < A), show that A≤1-β/α and B≥β/1-α

### Explanation

Let X= (X _{1}, …, X _{k}) and also let E _{k} be the set of all points in k dimensional Euclidean space R _{k}, for which we reject H _{0} using the sequential probability ratio test. Also, let F _{k } be the set of all points in R _{k} for which we accept H _{0}. Notice that (E _{k}, k = 1,2, …) are mutually disjoint and (F _{k}, k = 1,2, …) are also mutually disjoint. Assu

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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

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