Question

Need help simplifying this.

ln(product notation i =1 to n of (1/sigma(sqrt(2pi)))(e^-1/2((x-mu)/sigma)^2)))

n,pi,mu, and sigma are constants.

Answer 1

\[\Large \ln{\prod_{i=1}^n \frac{1}{\sigma\sqrt{2\pi}}} e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2}\]
If this is a question to look for maximum likelihood estimators, though, then this isn't the right expression to evaluate.

Answer 2

All I need is to simplify it using properties of logs, etc. x should be x subscript i. I'm just getting confused in trying to simplify it.

Answer 3

I see.
Then let us begin by simplifying the product.
Here are properties you can use:
\[\prod_{i=1}^n c=c^n \text{ where c is a constant}\]
\[\prod_{i=1}^n a^{x_i}=a^{\sum_{i=1}^nx_i} \text{where a is a constant}\]
Use these to simplify the product and let me know what you get.

Answer 4

Or use:
\[\ln \left( \prod_{i}f(x_i)\right)=\sum_{i}\ln(f(x_i))\]
Using that:
\[\ln(ab) = \ln(a) + \ln(b)\]

Answer 5

\[\ln((1/(\sigma(\sqrt(2\pi))))) + -(1/2)\sum_{i=1}^{n}((x _{i}-\mu)/\sigma)^2)\]

Answer 6

Correct?