ISS Previous year 2016 paper-02 Q. No. 2 Solution

ISS Previous year 2016 paper-02 Q. No. 2 Solution

ISS Previous Year Question – Quadratic Form Distribution

Previous year questions are extremely useful for understanding the conceptual level of the UPSC Indian Statistical Service (ISS) examination. The following question is based on the distribution of a quadratic form involving a normal vector and an idempotent matrix.

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Question 2

Let yi ~ N(0,1) where i = 1, 2, 3, …, n be n independent standard normal variables.

Let y = (y1, y2, …, yn)' be the vector of these variables.

Suppose y' A y is a quadratic form where A is an idempotent matrix.

What will be the distribution of y' A y?

Options:

(a) Normal distribution (b) Chi-square distribution (c) Bivariate Normal distribution (d) Rectangular distribution

Concept Used

If y ~ N(0, In) and A is an idempotent matrix, meaning A² = A then the quadratic form

y' A y follows a Chi-square distribution.

More precisely, y' A y ~ Chi-square distribution with r degrees of freedom where r = rank(A).

Explanation

Given that

yi ~ N(0,1) and all variables are independent, the vector y = (y1, y2, ..., yn)' follows a multivariate normal distribution y ~ N(0, I_n) where I_n is the identity matrix.

Now consider the quadratic form y' A y If matrix A satisfies A² = A then A is called an idempotent matrix.

A very important result in multivariate statistics states that:

• If y ~ N(0, I_n) and A is idempotent with rank r, then y' A y ~ Chi-square distribution with r degrees of freedom.

Final Answer

Therefore, y' A y follows a Chi-square distribution.

Correct option:

(b) Chi-square distribution

Exam Insight for ISS Aspirants

This is a very important theoretical result used in:

• Linear Models • Analysis of Variance (ANOVA) • Quadratic forms in normal variables • Statistical inference. In many ISS and statistics exams, whenever you see:

y' A y with y ~ N(0, I) • A idempotent you should immediately think: Chi-square distribution with rank(A) degrees of freedom.

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