If Xi ~ N(0,1), derive the probability density function of Xi^2. Write down the probability density function of SigmaXi^2

>>Jacques

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- Mar 20th 2008, 04:52 AMJacquesRouxDistribution Theory
If Xi ~ N(0,1), derive the probability density function of Xi^2. Write down the probability density function of SigmaXi^2

>>Jacques - Mar 20th 2008, 05:02 AMmr fantastic
I don't have the time right now to post a solution. But if you read number 10 of this you just might figure it out yourself ....

If you're still stuck, please say where - if no-one else replies I'll post some help in the morning (*my*morning!) - Mar 20th 2008, 08:25 AMJacquesRoux
Thanks very much will let you know if I need more help!

- Mar 23rd 2008, 10:39 AMJacquesRouxDistribution Theory
C = Integral R exp(-z2 / 2)dz = (2 )1/2.

Hint: Express C^2 as a double integral over R^2 and then convert to polar coordinates:

Is this the correct method for the above problem. If it is the correct one how will I determine the intervals of the double integral. I am really stuck with this problem and really will appreciate guidedance with this problem. (Headbang) Please help me!! - Mar 23rd 2008, 02:40 PMmr fantastic
I'm sorry but

C = Integral R exp(-z2 / 2)dz = (2 )1/2.

Hint: Express C^2 as a double integral over R^2 and then convert to polar coordinates:

is unclear to me. What is R? Does exp(-z2 / 2) mean ? What does (2 )1/2 mean? It would help if you could typeset these sorts of complex expressions using latex. Where has the hint come from? - Mar 23rd 2008, 03:10 PMJacquesRouxDistribution Theory
Sorry this was the question:

If Xi ~ N(0,1), derive the probability density function of Xi^2. Write down the probability density function of SigmaXi^2

and I have no clue how to even start the problem. Can you please give me some guidedance on how to even begin solving this problem.

Jacques - Mar 23rd 2008, 04:19 PMmr fantastic
Let

.

Therefore ,

where the Fundamental Theorem of Calculus and the chain rule have been used to get the derivative

,

and f(x) = 0 for x < 0.

As expected, this is the pdf for the chi-square distribution of degree 1. Note: .

You could also find the moment generating function of and recognise it as being the moment generating function of the chi-square distibution of degree 1. - Mar 24th 2008, 02:13 AMJacquesRouxDistribution Theory
Thanks very much!!

Jacques(Rock)