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

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anonymous
  • anonymous
Or use: \[\ln \left( \prod_{i}f(x_i)\right)=\sum_{i}\ln(f(x_i))\] Using that: \[\ln(ab) = \ln(a) + \ln(b)\]
anonymous
  • anonymous
\[\ln((1/(\sigma(\sqrt(2\pi))))) + -(1/2)\sum_{i=1}^{n}((x _{i}-\mu)/\sigma)^2)\]
anonymous
  • anonymous
Correct?

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