# Math Help - Distribution Theory

1. ## Distribution Theory

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

2. Originally Posted by JacquesRoux
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!)

3. Thanks very much will let you know if I need more help!

4. ## Distribution 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. Please help me!!

5. Originally Posted by JacquesRoux
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 $e^{-z^2/2}\,$ ? 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?

6. ## Distribution 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

7. Originally Posted by JacquesRoux
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 $F(x) = \Pr(\chi^2 < x)$

$\Rightarrow F(x) = \Pr(-\sqrt{x} < \chi < \sqrt{x})$

$= \frac{1}{\sqrt{2 \pi}} \, \int_{-\sqrt{x}}^{\sqrt{x}} e^{-u^2/2} \, du$

$= \frac{2}{\sqrt{2 \pi}} \, \int_{0}^{\sqrt{x}} e^{-u^2/2} \, du$.

Therefore $f(x) = \frac{dF}{dx} = \frac{2}{\sqrt{2 \pi}} \, e^{-x/2} \, \left (\frac{1}{2 \sqrt{x}} \right) \,$ , $x > 0$

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

$= \frac{1}{\sqrt{2 \pi}} \, e^{-x/2} \, \frac{1}{\sqrt{x}} \,$ , $x > 0$

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

As expected, this is the pdf for the chi-square distribution of degree 1. Note: $\frac{1}{\sqrt{2 \pi}} = \frac{1}{\Gamma(1/2) 2^{1/2}}$.

You could also find the moment generating function of $\chi^2$ and recognise it as being the moment generating function of the chi-square distibution of degree 1.

8. ## Distribution Theory

Thanks very much!!

Jacques