If Xi ~ N(0,1), derive the probability density function of Xi^2. Write down the probability density function of SigmaXi^2
>>Jacques
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!)
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. Please help me!!
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?
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
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.