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Mathematical Statistics with Applications | 7th Edition | ISBN: 9780495110811 | Authors: Dennis Wackerly; William Mendenhall; Richard L. Scheaffer
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Suppose that W = Y1 + Y2 where Y1 and Y2 are independent.

Chapter 6, Problem 60E

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QUESTION:

Problem 60E

Suppose that W = Y1 + Y2 where Y1 and Y2 are independent. If W has a χ 2 distribution with νdegrees of freedom and W1 has a χ 2 distribution with ν1 < ν degrees of freedom, show that Y2 has a χ 2 distribution with ν − ν1 degrees of freedom.

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QUESTION:

Problem 60E

Suppose that W = Y1 + Y2 where Y1 and Y2 are independent. If W has a χ 2 distribution with νdegrees of freedom and W1 has a χ 2 distribution with ν1 < ν degrees of freedom, show that Y2 has a χ 2 distribution with ν − ν1 degrees of freedom.

ANSWER:

Solution

Step 1 of 2

We have to show that Y2 has a distribution with degrees of freedom

Given that Y1 and Y2 are independent random variables

And W=Y1 + Y2 

Here W has a distribution with degrees of freedom

And  Y1 has a distribution with degrees of freedom

The moment generating function of X is

The moment generating function of W has a distribution  with degrees of freedom is

                                     

The moment generating function of  Y1 has a distribution  with degrees of freedom is

                                      

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