Simplify this expression with Mathematica

[dabd]

I know this expression should return a Gaussian distribution but I can't get Mathematica to simplify the integral. What am I missing?

$$\text{Simplify}\left[\frac{\text{Product}\left[\text{PDF}\left[\text{NormalDistribution}[y,\sigma ],x_i\right],\{i,n\}\right]\text{PDF}[\text{NormalDistribution}[\mu ,\phi ],y]}{\text{Integrate}\left[\text{Product}\left[\text{PDF}\left[\text{NormalDistribution}[y,\sigma ],x_i\right],\{i,n\}\right]\text{PDF}[\text{NormalDistribution}[\mu ,\phi ],y],\{y,-\text{Infinity},\text{Infinity}\},\text{Assumptions}\to \{\sigma >0,\phi >0\}\right]},\{\sigma >0,\phi >0\}\right]$$

 

[jackmell]

Kinda' messy dab. Runs right off the right side of my computer. Try FullSimplify and just on the part you need to simplify. Then start adding things back to the expression one by one to see if that helps.

 

[dabd]

Ok, so I am trying to integrate this with no success

FullSimplify[
Integrate[
Product[PDF[NormalDistribution[y, \[Sigma]], Subscript[x, i]], {i,
n}]

PDF[NormalDistribution[\[Mu], \[Phi]], y],
{y, -Infinity, Infinity},
Assumptions -> {\[Sigma] >= 0, \[Phi] >= 0}]]

 

[Hepth]

Im not sure what "n" is, but if its 1, what you need to do is change the greater than equals signs to just greater than signs, I get:

n = 1
Integrate[
  Product[PDF[NormalDistribution[y, \[Sigma]], Subscript[x, i]], {i, 
     n}] PDF[NormalDistribution[\[Mu], \[Phi]], y], {y, -Infinity, 
   Infinity}, 
  Assumptions -> {\[Sigma] > 0, \[Phi] > 0}] // FullSimplify

$$\frac{e^{-\frac{\left(\mu -x_1\right){}^2}{2 \left(\sigma ^2+\phi ^2\right)}}}{\sqrt{2 \pi } \sqrt{\sigma ^2+\phi ^2}}$$

 

[dabd]

By letting n=1 you eliminate the product and that is not what I meant.
n is simply the upper limit of the product, i.e., 'i' goes from 1 to 'n'.

Thanks.