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If I have two random variables as follows:

1) A Gaussian distribution of wifi signal strengths at a known point

2) A Gaussian distribution of wifi signal strengths at an unknown point

(Note that the above readings are determined using the same routers (access points).)

When trying to create a joint probability distribution, would we consider these two variables to be statistically independent and hence use "Sum of normally distributed random variables"?

http://en.wikipedia.org/wiki/Sum_of_normally_distributed_random_variables

It seems that I need to determine statistical independence to decide on how to work out the joint probability distribution. However it also seems that I need to work out the joint probability distribution to determine statistical independence.

I don't understand how I'm meant to determine whether my variables are independent or not?

http://en.wikipedia.org/wiki/Statistical_independence#For_events

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    $\begingroup$ In my (limited) experience, there are two routes to go by: 1) If you assume the variables are jointly normally distributed, then do a test for the correlation coefficient to be 0. That would determine independence. 2) If you aren't willing to assume that, then usually independence is asserted (or not) based on experimental conditions (is there any possible reason to believe those signal strengths might have dependency?). This is a "soft" approach though, no formal testing. $\endgroup$
    – ved
    Jul 1, 2014 at 3:56
  • $\begingroup$ "It seems that I need to determine statistical independence to decide on how to work out the joint probability distribution.' Yes. "However it also seems that I need to work out the joint probability distribution to determine statistical independence." No. What gave you this idea? $\endgroup$
    – Did
    Jul 4, 2014 at 5:41

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Given your problem as stated, it appears that assuming independence would be incorrect, since you don't know if the unknown location is essentially right next to your known location or on the other side of the globe. Since they are Gaussain, you can model the sum as a sum of correlated gaussian random variables, in which case, inference on the correlation coefficient would give you some insight into how close the two sources are to each other.

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