No.
If the two random variables X and Y are independent, then the pdf of Z = XY is probably (I haven't done the calculation) a Bessel function. See 3. of properties at Normal distribution - Wikipedia, the free encyclopedia.
Hello,
I am trying to find the distribution of the product of two normal densities (different means and standard deviations) . Does the product follow normal distribution also? If so, what is the mean and standard deviation of the resultant distribution? Thanks a lot.
No.
If the two random variables X and Y are independent, then the pdf of Z = XY is probably (I haven't done the calculation) a Bessel function. See 3. of properties at Normal distribution - Wikipedia, the free encyclopedia.
I have the following two densities:
f1(X) = {1/sigma1*sqrt(2*pi}* exp{-(X-mu1^2)/2*sigma1^2}
f2(X) = {1/sigma2*sqrt(2*pi}* exp{-(X-mu2^2)/2* sigma2^2}
So f1 is N(mu1, Sigma1^2) and f2 is N(mu2, Sigma2^2).
Does f(x) = f1(X)*f2(X) follow a normal distribution? Thanks a lot.
In Page 3 of the following link, it is given that the product of two gaussian densities is also gaussian. I think it is same as the one I requested above. I am looking for the derivation of the formula given for the resultant distribution:
http://www.cs.cmu.edu/afs/cs/academi...ww/hw5/hw5.pdf
confirmed here as well:
Product of Two Gaussian PDFs
If I understand you correctly, what you're asking boils down to wanting to show that
and getting the appropriate expressions for and C in terms of and . Note that becomes part of the normalising constant.
It's simple to show and get the expressions but tedious to type out.
That f(x) is of the form of a Gaussian follows from the convolution theorem and the fact that the Fourier transform of a Gaussian is a Gaussian. But this does not guarantee that the product of the densities is in fact a density. As it happens it would be quite supprising if it were.
So we can bust a gut attempting to prove that , but we can just do the experiment and evaluate the thing numerically. The answer is that the integral is not so is not a density.
RonLCode:>s1=1,mu1=0,s2=2, mu2=5 1 0 2 5 >dx=0.2; >x=-10+dx/2:dx:20; > >f1=1/(s1*sqrt(2*pi))* exp( -(x-mu1)^2 / (2*s1^2) ); >f2=1/(s2*sqrt(2*pi))* exp( -(x-mu2)^2/ (2*s2^2) ); > >f=f1*f2; > >II1=sum(f1)*dx 1 >II2=sum(f2)*dx 1 >II=sum(f)*dx 0.014645 >
I have a couple of more questions on this product of two distributions:
1. In the derivation given by mr. fantastic, constant C has been ignored why defining mu and sigma? I am also not sure how can we define mu and sigma if the resultant expression is not a pdf?
2. Is this extensible for the multivariate case? I would appreciate if you can provide me some references in this regard.
Thanks a lot.
Hi I have a problem similar to the poster's
f1(X) = {1/sigma1*sqrt(2*pi}* exp{-( y1-mu1^2)/2*sigma1^2}
f2(X) = {1/sigma2*sqrt(2*pi}* exp{-( y2-mu2^2)/2* sigma2^2}
notice the y1 and y2
and now y1 = x + n1
and y2 = x + n2
and n1 and n2 are normal distributions
I need to find the product of two functions and prove x follows a normal distribution...
Thank you.