Reciprocal of Normal Distribution
shafriza
I found out this forum and I wish I could get come advice. Let say I
have a number X with normal distribution, and I want to find out what
is the the distribution of Z=3/X. Appreciate your advice. I tried to
plot using Matlab and I found out its not normal and skewed to the
center.
Thank you,
shafriza
Roland Franzius
The ratio of two independent normal distributed random variables has a
Cauchy distribution. "3" and "X" are independent and normal distributed
random variables.
See eg Wikipedia, Mathworld or eom.springer.de
--
Roland Franzius
Robert Israel
The distribution of Z can be obtained by standard techniques, found in
most undergraduate probability textbooks. The density is
f(z) = 3 exp(-9/(2 z^2))/(sqrt(2*pi) |z|^2) for z <> 0
The cumulative distribution function is
F(z) = -1/2 erf(3/(sqrt(2) z)) for z < 0,
1/2 for z = 0
1 - 1/2 erf(3/(sqrt(2) z)) for z > 0
--
Robert Israel isr...@math.MyUniversitysInitials.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada
Robert Israel
The ratio of two normally distributed random variables with mean 0 has a
Cauchy distribution. X might have mean 0 (I see that, without justification,
I assumed in my posting that X was "standard normal", i.e. mean 0 and variance
1). The constant 3, even if you consider it as a degenerate case of a normal
distribution, certainly does not have mean 0. So 3/X does not have a
Cauchy distribution.
Ian Parker
does indeed produce another normal distribution. However take
d(x) = dx*exp(-x*x) (Ignore constant factors)
put z=1/x now dz=-dx/x*x so that dx = -dz/z*z giving
d(z) = -exp(-1/z*z)/z*z or exp(-1/z*z)/z*z removing the - sign. This
is far from normal. MATLAB is probably correct in its calculation. How
far away from the origin were you?
- Ian Parker
Roland Franzius
> Roland Franzius <roland....@uos.de> writes:
>
>> shafriza schrieb:
>>> Hi all,
>>>
>>> I found out this forum and I wish I could get come advice. Let say I
>>> have a number X with normal distribution, and I want to find out what
>>> is the the distribution of Z=3/X. Appreciate your advice. I tried to
>>> plot using Matlab and I found out its not normal and skewed to the
>>> center.
>> The ratio of two independent normal distributed random variables has a
>> Cauchy distribution. "3" and "X" are independent and normal distributed
>> random variables.
>>
>> See eg Wikipedia, Mathworld or eom.springer.de
>
> The ratio of two normally distributed random variables with mean 0 has a
> Cauchy distribution. X might have mean 0 (I see that, without justification,
> I assumed in my posting that X was "standard normal", i.e. mean 0 and variance
> 1). The constant 3, even if you consider it as a degenerate case of a normal
> distribution, certainly does not have mean 0. So 3/X does not have a
> Cauchy distribution.
Sorry to OP, you are right of course. Overlooked the normalization
condition in wiki.
The distribution of the inverse of random variable with distribution F
is expressible by the original distribution.
From
Pr[X < x] = F(x)
one has
Pr[3/X <x] =
x<0: Pr[X<0 && X>3/x || X>0 && X<3/x] = F(0)-F(3/x)
x>0: Pr[X<0 && X< 3/x || X>0 && X> 3/x ] = F(0) + 1-F(3/x)
For F continous, this is a continous ditribtion function increasing from
0 to 1 with density
Pr [3/X \in (x,x+dx)] = 3/x^2 F'(3/x) dx + o(dx^2)
The general result for the inverse Gaussian
Pr [3/X \in (x,x+dx)]
= 3 dx/(sqrt (2 pi) x^2 sigma) exp(- (3/x-m)^2/(2 sigma^2))
At x=0, the density is zero to all orders.
--
Roland Franzius