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I know that two normal distributions can be subtracted and get a new distribution with a mean of $\bar{x} = \bar{x}_1-\bar{x}_2$ and a standard deviation of $\sigma = \sqrt{\sigma_1^2 + \sigma_2^2}$. Does someone know if the same applies for the subtraction of two skewed (in my case Gumbel) distributions, so if the new distribution's mean and std are connected with the initial ones with the same relations?

Glen_b
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Chris
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    1. You're subtracting random variables, not subtracting their distributions. 2. You need to specify something about the dependence (enough to infer the correlation) to compute the variance of the difference. 3. You appear to be confusing the means of the normal distributions ($\mu_1,\mu_2$) with their sample means ($\bar{x}_1,\bar{x}_2$). – Glen_b Sep 28 '17 at 11:09
  • Partial duplicate [here](https://stats.stackexchange.com/questions/142745/what-is-the-demonstration-of-the-variance-of-the-difference-of-two-dependent-var/) – Glen_b Sep 28 '17 at 13:45

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Where the population means, and variances are finite respectively, we can use basic properties of means and variances:

$\mu=E(X_1-X_2)=E(X_1)-E(X_2)=\mu_1-\mu_2$

For uncorrelated random variables: $\text{Var}(X_1-X_2)=\text{Var}(X_1)+\text{Var}(X_2)$.

More generally: $\text{Var}(X_1-X_2)=\text{Var}(X_1)-2\text{Cov}(X_1,X_2)+\text{Var}(X_2)$.

None of these results depend on the distribution.

[However, it is usually not the case that the distribution of a difference is in the same family of distributions as the original variables.]

Glen_b
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  • The mean of a Gumbel distribution is $\mu+\beta*\gamma$ where $\mu$ and $\beta$ are the location and scale parameter, and $\gamma$ the Euler constant. Is this the population mean? "You're subtracting random variables, not subtracting their distributions" Do you mean the means as the variables? I.e. my distributions are describing present and future maxima of a magnitude, wouldn't $\mu$ and $Var(X1-X2)$ describe their difference? – Chris Sep 28 '17 at 14:00
  • 1. Yes, the expectation of the variable is the population mean. 2. No. Note that $X_1$ and $X_2$ are random variables. You want information about the distribution of the difference of those variates, $X_1-X_2$, not the difference in their distributions $F_{X_1}(x)-F_{X_2}(x)$ (which isn't itself a distribution; or if you intend 'distribution' to refer to densities, the difference in those is not a density either). Your title and body should correctly describe what you seek. 3. I'm not quite sure what you're saying there in the last bit; certainly mean and variance tell you something. – Glen_b Sep 28 '17 at 15:46
  • I see. When the mean and the variance though refer to two distributions(which describe different time periods of the same variable), shouldn't $\mu$ and $Var(X_1-X_2)$ contain some information about the difference of these two distributions? – Chris Sep 29 '17 at 12:05
  • Yes, they contain information about the distribution of the difference just as the mean and variance contain some information about any distribution (but again, don't persist in calling the distribution of the difference a difference in distributions) – Glen_b Sep 29 '17 at 13:34