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# Probability distribution and variance of the sum of two normally distributed random variables.

Posted on 2011-03-03
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What is the shape of the probability distribution and the variance of random variable C,
where
C = A + B,
A and B are normally distributed variables

if
- The mean of A is 0, and the mean of B is 0.
- The variance of A is a, and the variance of B is b.

?
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Question by:Milewskp
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Accepted Solution

TommySzalapski earned 500 total points
http://en.wikipedia.org/wiki/Sum_of_normally_distributed_random_variables
The sum of two normal random variables is a normal distribution.
The mean is the sum of the means and the variance is the sum of the two variances.
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Expert Comment

Any more than that and I'd have to do your homework for you! Once you know the rule, it's very simple.
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Expert Comment

I think there is a difficulty with
"The mean is the sum of the means"
If you have a distribution with mean 1 and one with mean 2 the mean of the sum cannot be 3
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Expert Comment

Yes it can and yes it is. It's normal distributions, why wouldn't it be 3?
If the mean of A is 1 and the mean of B is 2 and the are notmal variables, then C is normal and the mean of C is 3.
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Expert Comment

"Yes it can and yes it is"
One should be very careful about taking my not-to-well informed opinion over TS but consider two populations with VERY small SD (and equal sizes). One has a mean of 1 and one a mean of 2 (with SD of 0.00001.
I find it hard to see that the mean of the combined distribution would be other than 1.5. Another big difficulty would be the fact that in the combined distribution there would be many more measurments with a value <3 than >3.
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I looked at your link and you definitely agree with it. I must be misunderstanding a definition someplace. I may be thinking too experimentally as it should be clear that my two example populations are not samples of the same population. but I am still puzzled.
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It's based totally on the fact that the problem said they were normal distributions. It doesn't matter what the population size is if you know the distribution.
You can't really be 100% positive that you know the distribution if you are doing experiments, but if the homework problem says normal distribution, then you know you can add them like that.
It's a property of the pdf of the normal distribution.
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Expert Comment

TS
you are right and I certainly cannot argue with the math in your link, but there is more than meets my eye in the language. But to make sure that the asker does not misunderstand anything you are right.
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