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I am trying to determine if these two expressions are equivalent:

$$\prod_{i=1}^n\frac{\log(\theta)}{\theta - 1} \theta^{x_i} = \frac{(\log(\theta))^n}{(\theta - 1)^n}\theta^{\sum_{i=1}^n x_i}$$

If they aren't what am I missing here?

statrat403
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    Yes, they are equivalent. You should, as @ChrisHaug requests, be more precise with your indices as a general rule, in this case by including them in your product and sum terms; in this case we can deduce what you mean, but in other cases we might not be able to - or, worse yet, someone like me who tends to guess (too often) might guess wrong and waste a lot of everyone's time. – jbowman Mar 07 '18 at 18:43
  • sorry, I did not know how to add that in with formatting. The product and sum terms both go from i = 1 to n – statrat403 Mar 07 '18 at 18:47
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    This is simply a question of manipulating an algebraic expression, I'd suggest an immediate review of the basic rules of manipulating products and powers. – Glen_b Mar 08 '18 at 00:57

2 Answers2

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Assuming that theta is not defined in terms of i, then indeed they are correct: $$ \prod_i^N{k\theta^{x_i}} = k^N\prod_i^N{\theta^{x_i}}=k^N\theta^{\sum_i^N{x_i}} $$

Angus Hollands
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Note:

$$\prod_i^n\frac{log(\theta)}{\theta-1}\theta^{x_i}=\prod_i^n\frac{log(\theta)}{\theta-1}*\prod_i^n\theta^{x_i}=\left(\frac{log(\theta)}{\theta-1}\right)^n\theta^{\sum_i^n x_i}$$

Hope this is helpful!

Bensstats
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