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Lets say I simulate $X,Y$ and $Z$ from 3 standard normal distributions with pairwise correlations of 0.5, then I calculate the sum $X+Y-Z$. According to https://en.wikipedia.org/wiki/Sum_of_normally_distributed_random_variables#Correlated_random_variables, since $X,Y,Z$ are jointly normally distributed, it seems the sum will be another normally distributed random variable with $\mu = 0$ and

$$ \sigma = \sqrt{ \sigma_X^2 + \sigma_Y^2 + \sigma_Z^2 + 2\sigma_{XY} - 2\sigma_{XZ} - 2\sigma_{YZ} } $$

If, alternatively, $X,Y$ and $Z$ have the same correlation but are not jointly distributed, how does the distribution of $X+Y-Z$ change?

Furthermore, how would I even go about simulating $X,Y,Z$ with pairwise correlations of 0.5 that are not jointly distributed?

stav
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  • What you claim is the standard deviation of $X+Y-Z$ is wrong; two $+$ signs should be $-$ signs. So either you didn't correctly copy what Wikipedia says is the standard deviation of $X+Y-Z$ or you copied correctly and it is the Wikipedia page that has typographic errors in it. Which is it? – Dilip Sarwate Apr 12 '20 at 21:10
  • I've edited the question to make it more clear. – stav Apr 12 '20 at 21:31
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    See https://stats.stackexchange.com/a/30205/919 for a great discussion of the central question about Normal variables that are not jointly Normal. – whuber Apr 12 '20 at 21:40

1 Answers1

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Jointly normal RVs have a known form, but others don't. So, if a pair or triple of RVs are not jointly distributed, you have to know how they're connected to each other, which is specified via their specific joint distribution. For example, the following figure in this wiki page shows two different non-normal joint distributions' scatter plots (that have normal marginals):

enter image description here

Once you have the joint distribution and how the variables are related, your sampling strategy will depend on it.

gunes
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