Thread: Really confusing distribution problem: standard normals

1. Really confusing distribution problem: standard normals

If U is uniform on (0,2pi) and Z, independent of U, is exponential with rate 1, show directly that X and Y definited by:

X=sqrt(2Z)*cosU
Y=sqrt(2Z)*sinU
are independent standard normal random variables.

I have no idea how to even start this one besides just writing the functions for U and Z....

2. Hello,
Originally Posted by zhupolongjoe
If U is uniform on (0,2pi) and Z, independent of U, is exponential with rate 1, show directly that X and Y definited by:

X=sqrt(2Z)*cosU
Y=sqrt(2Z)*sinU
are independent standard normal random variables.

I have no idea how to even start this one besides just writing the functions for U and Z....
See here : Box?Muller transform - Wikipedia, the free encyclopedia

The derivation[3] is based on the fact that, in a two-dimensional cartesian system where X and Y coordinates are described by two independent and normally distributed random variables, the random variables for R2 and Θ (shown above) in the corresponding polar coordinates are also independent and can be expressed as

$R^2 = -2\cdot\ln U_1\,$

and

$\Theta = 2\pi U_2.\,$
fyi, -ln(uniform distribution over (0,1)) follows an exponential distribution with parameter 1. It can be proved by using this : http://en.wikipedia.org/wiki/Inverse_transform_sampling

3. Thanks, even though it doesn't exactly help since it just presents a more muddled version of what the textbook presents.

4. Originally Posted by zhupolongjoe
Thanks, even though it doesn't exactly help since it just presents a more muddled version of what the textbook presents.
Huh ?
Oh no, I'm sorry I completely misread the question

I don't really know an "immediate" way... But you can find the joint pdf, and then make a 2-2 transform (as matheagle calls it).
That is a change of variables (namely, changing into polar coordinates)

5. Okay, I know how to compute the Jacobian, but I am unsure about the joint pdf...I can't figure out exactly what it is I should integrate over....\

EDIT

disregard...I solved it, thanks though

6. Originally Posted by zhupolongjoe
Okay, I know how to compute the Jacobian, but I am unsure about the joint pdf...I can't figure out exactly what it is I should integrate over....\

EDIT

disregard...I solved it, thanks though
In that case, you might consider posting your solution so that others might benefit from it (or point out mistakes in it .... )

7. Originally Posted by Moo
Huh ?
Oh no, I'm sorry I completely misread the question

I don't really know an "immediate" way... But you can find the joint pdf, and then make a 2-2 transform (as matheagle calls it).
That is a change of variables (namely, changing into polar coordinates)
Yes, it's named after me.
Mathbeagle 2-2=0 transformation, lol.
Eye no I'm in trouble when Moo 'quotes' me

8. Ok here is the solution I did...

J(R^2,theta)= det(2x 2y; -y/(x^2+y^2) x/(x^2+y^2)
=2

Set R^2=2Z
f(d,theta)=(1/2)*e^(-d/2)*(1/2pi)
0<d<infinity and 0<theta<2pi
So R^2 and theta are independent

We have theta=U and X1=Rcos theta, Y=Rsin theta
R^2 is chi-squared with 2 degrees of freedom and sowe have from these things

X=sqrt(2Z) cos U and Y=sqrt(2Z)sin U and they are independent, as we needed

,
,

If U is uniform on (0, 2π) and Z, independent of U, is exponential with rate 1, show directly (without using the results of Example 7b) that X and Y defined by X = 2ZcosU √ √ Y = 2ZsinU are independent standard normal

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