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glare22@gmail.com science forum beginner
Joined: 12 Jul 2006
Posts: 7

Posted: Thu Jul 13, 2006 9:08 am Post subject:
How to get the analytical PDF of this expression?



Can anyone tell me the pdf of this expression u0+u1*x+u2(x^21)?
where u0,u1,u2 is constant. And x is a random variable satisfing the
normal distribution N(0,1). Thanks a lot. 

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Torsten Hennig science forum Guru Wannabe
Joined: 28 Apr 2005
Posts: 136

Posted: Thu Jul 13, 2006 10:46 am Post subject:
Re: How to get the analytical PDF of this expression?



Quote:  Can anyone tell me the pdf of this expression u0+u1*x+u2>(x^21)?
where u0,u1,u2 is constant. And x is a random variable >satisfing the
normal distribution N(0,1). Thanks a lot.

Hi,
let Y = u0 + u1*X + u2*(X^21).
For a in IR, calculate the Xvalues for which Y<=a.
(So solve u0 + u1*X + u2*(X^21) <= a for X).
Let the corresponding intervall be given by
[l(a),u(a)].
Then the pdf for Y , f_Y, is given by
f_Y(a) = f_X(u(a))*u'(a)  f_X(l(a))*l'(a).
Try the formula for Y = X^2.
You should arrive at
f_Y(a)=1/(2*sqrt(a))*(f_X(sqrt(a))+f_X(sqrt(a))) (a>0).
Best wishes
Torsten. 

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Duncan Muirhead science forum addict
Joined: 08 Oct 2005
Posts: 70

Posted: Thu Jul 13, 2006 11:57 am Post subject:
Re: How to get the analytical PDF of this expression?



On Thu, 13 Jul 2006 02:08:33 0700, glare22 wrote:
Quote:  Can anyone tell me the pdf of this expression u0+u1*x+u2(x^21)?
where u0,u1,u2 is constant. And x is a random variable satisfing the
normal distribution N(0,1). Thanks a lot.
If we assume u2>0 then 
u0 + u1*x + u2*(x*x1) = u2*y^2 + u0  u1*u1/(4*u2)  u2
where y = xu1/(2*u2) is a N(u1/(2*u2), 1) variable.
y^2 has a noncentral chisquare distribution with 1
degree of freedom, whose distribution function you can
find by searching the web (for noncentral chisquare),
and then a linear transformation will get the distribution
of the variable you want.
Duncan 

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glare22@gmail.com science forum beginner
Joined: 12 Jul 2006
Posts: 7

Posted: Fri Jul 14, 2006 11:46 pm Post subject:
Re: How to get the analytical PDF of this expression?



Thank Torsten Hennig and Duncan Muirhead a lot.
Duncan Muirhead wrote:
Quote:  On Thu, 13 Jul 2006 02:08:33 0700, glare22 wrote:
Can anyone tell me the pdf of this expression u0+u1*x+u2(x^21)?
where u0,u1,u2 is constant. And x is a random variable satisfing the
normal distribution N(0,1). Thanks a lot.
If we assume u2>0 then
u0 + u1*x + u2*(x*x1) = u2*y^2 + u0  u1*u1/(4*u2)  u2
where y = xu1/(2*u2) is a N(u1/(2*u2), 1) variable.
y^2 has a noncentral chisquare distribution with 1
degree of freedom, whose distribution function you can
find by searching the web (for noncentral chisquare),
and then a linear transformation will get the distribution
of the variable you want.
Duncan 


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