I was reading a book and came across with a equation which gives the normal distribution function of continuous random variable. It was used in a software called RapidMiner to visualize data distribution.
$f\left( x \right) = \frac{1}{{\sqrt {2\pi \sigma } }}{e^{\frac{{{{\left( {x - \mu } \right)}^2}}}{{2{\sigma ^2}}}}}$ , where $x$ is random variable, $\sigma$ is the standard deviation, and $\mu$ is the mean of the distribution.
I got badly confused, as it is different and not same as what I learnt from textbook or wikipedia, which is:
$f\left( x \right) = \frac{1}{{\sigma \sqrt {2\pi } }}{e^{ - \frac{{{{\left( {x - \mu } \right)}^2}}}{{2{\sigma ^2}}}}}$ where the symbols mean the same.
I have questions:
- is the equation in the book correct?
- if so, could you please walk me through how could the equation be correct? Maybe give me some keyword for searching.
- if wrong, could you tell me why?
Finally the book luckily provide a on-line reading site: http://www.learnpredictiveanalytics.com/preview.html . The context of the equation of my question is on Page 51, right below figure 3.9.
Thank you.
Deal all, I just noticed someone answered my question but soon deleted. She/he mentioned that performing integral calculation will tell the differernce. And actually it works and I realized and strongly believed that the equation on the book was wrong.
I input the equations to an online integral calculator (well, I know it was an bad idea). And found the result is "The integral is divergent" while the second one yields "1".
Problem solved.