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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^2-1)?
where u0,u1,u2 is constant. And x is a random variable satisfing the
normal distribution N(0,1). Thanks a lot.
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^2-1)? 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^2-1).
For a in IR, calculate the X-values for which Y<=a.
(So solve u0 + u1*X + u2*(X^2-1) <= 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.

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^2-1)? 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*x-1) = u2*y^2 + u0 - u1*u1/(4*u2) - u2
where y = x-u1/(2*u2) is a N(-u1/(2*u2), 1) variable.
y^2 has a non-central chi-square distribution with 1
degree of freedom, whose distribution function you can
find by searching the web (for non-central chi-square),
and then a linear transformation will get the distribution
of the variable you want.
Duncan
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.

 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^2-1)? 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*x-1) = u2*y^2 + u0 - u1*u1/(4*u2) - u2 where y = x-u1/(2*u2) is a N(-u1/(2*u2), 1) variable. y^2 has a non-central chi-square distribution with 1 degree of freedom, whose distribution function you can find by searching the web (for non-central chi-square), and then a linear transformation will get the distribution of the variable you want. Duncan

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