ISS 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

Essay Question▾

Describe in Detail

Let X= (X 1, X 2, X 3) ’ be distributed as N 3 (µ, ∑) where µ’= (2, -3, 1) and

Equation

(i) Find the distribution of 3X 1 -2X 2 +X 3.

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

Explanation

(i) the distribution of 3X 1 -2X 2 +X 3 is

Equation

The mean is

Equation

Equation

The variance-covariance matrix is

Equation

Equation

Equation

Equation

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

(ii) X 2 and Equation are independent if

Equation

Equation

Equation

Equation

Equation

Equation

Question number: 26

» Statistical Quality Control » Sequential Sampling Plans

Appeared in Year: 2015

Essay Question▾

Describe in Detail

To test the hypothesis Equation against Equation for the distribution

Equation

Develop the sequential probability ratio test.

Explanation

Let x 1, x 2, …, x n are independently and identically distributed with distribution is

Equation

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… (91 more words) …

Question number: 27

» Multivariate Analysis » Multivariate Normal Distribution » Partial Correlation Coefficient

Appeared in Year: 2015

Essay Question▾

Describe in Detail

Let X = (X 1, X 2, X 3) ’ be distributed as N 3 (µ, ∑) where µ = (10, -7, 2) ’ and

Equation

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

Equation

where these terms are defined by this equation

Equation

Given that

Equation

So, the value of equation is

Equation

Equation

Equation

The partial correlation is

Equation

Equation

Equation

Question number: 28

» Multivariate Analysis » Estimation » Covariance Matrix

Appeared in Year: 2015

Essay Question▾

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.

Equation

from a bivariate normal distribution.

Explanation

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

Equation

Assume

Equation

So,

Equation

Equation

Equation

Equation

Equation

Question number: 29

» Estimation » Estimation Methods » Methods of Moments

Appeared in Year: 2015

Essay Question▾

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

Equation

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

Equation

Equation

The estimating equations are

Equation

Equation

The above equation for solving the parameter, we get

Equation

Equation

By method of moments, we see the observations

Equation

Equation<span class="more">… (6 more words) …</span>

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