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How are we going to know that a continuous random variable is a normal random variable? The definition I believe for a continuous r.v. to be a normal r.v. is that it's probability density function must be the pdf of the normal distribution. How are we going to check the pdf of a continuous r.v. and compare it with the normal distribution?

I really need the help. Thanks guys.

TRUSKI
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Isaac Newton
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  • What is your rv? – ə̷̶̸͇̘̜́̍͗̂̄︣͟ Oct 23 '17 at 09:58
  • Just some random continuous rv. I'm trying to find whether there exist a procedure that one must do in order to check whether a continuous rv is a normal rv or not. – Isaac Newton Oct 23 '17 at 10:00
  • It sounds like: how to find out that a piece of fruit is an apple? – drhab Oct 23 '17 at 10:00
  • Do you have an equation? Dataset? Without more information, not much to be done. Maybe graph a histogram of your data and fit it with the normal pdf. Try a statistical test, like KS-test, which compares the cdf. – jdods Oct 23 '17 at 10:00
  • Compare the pdf of your rv with $\frac{1}{\sigma \sqrt{2 \pi}} \textrm{exp}\left[-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2\right]$ – ə̷̶̸͇̘̜́̍͗̂̄︣͟ Oct 23 '17 at 10:03
  • @drhab Yup. That hits the mark. – Isaac Newton Oct 23 '17 at 10:04
  • @jdods All I have is the knowledge that it is a continuous rv. So do I need a data set? Should gathering some data be included in the procedure that I'm looking for? – Isaac Newton Oct 23 '17 at 10:07
  • @thesimplifire How am I going to produce the pdf of a continuous rv? – Isaac Newton Oct 23 '17 at 10:27
  • @Issac have you learnt about characteristic function? Some times the characteristic function are more easy to verify a random variable. See https://www.ma.utexas.edu/users/gordanz/notes/characteristic.pdf – TRUSKI Oct 23 '17 at 10:34
  • @truski Nope. I havent learned about it yet. Can I use the idea of a characteristic func to check whether its underlying pdf is the normal distribution? – Isaac Newton Oct 23 '17 at 10:55
  • http://www.cs.columbia.edu/~ccanonne/files/misc/2015-survey-distributions.pdf

    This document is a survey on 'testing' distributions. What is important is that it focuses on methods that have provable guarantees.

    What might be relevant to you is the chapters on testing closeness between distributions.

     – Abhiram Natarajan Oct 23 '17 at 10:57

1 Answers1

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If you have certain data regarding your specific random variable, one of the methods to deduce its distribution is the so called statistical inference, where you try to find the distribution (and respective pdf) that best fits your data. Sometimes you're only interested in some properties of the random variable, like its mean or variance, and not necessarily the distribution itself.

There are parametric and nonparametric approaches to this, you should check these definitions carefully.

sam wolfe
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