Sum of two identically distributed variables
We now concentrate on the sum of two independent random quantities with identical distributions — e.g. a random sample of size 2 from a distribution with mean µ and standard deviation σ. From the formulae on the previous page, this has mean and standard deviation:
Sum of two variables with different means
We now generalise by allowing X1 and X2 to have different means, µ1 and µ2, but the same standard deviation. Their sum has a distribution with the same spread as above, but the formula for the mean must be generalised:
Difference between two variables
A similar result holds for the difference between X1 and X2. If they both have standard deviation σ, their difference has the same standard deviation as their sum (but the distribution has a different mean):
Shape of distribution
If X1 and X2 are independent and have normal distributions, their sum and difference are also normally distributed.
If X1 and X2 have distributions with different shapes, their sum and difference usually have distributions that are non-normal but are closer to normal than the two source distributions. However the above formula for the mean and standard deviation hold whatever the shapes of the distributions of X1 and X2.
Illustration
The top of the diagram below shows the distributions of two normal variables, X1 and X2. The longer vertical red lines above each distribution can be dragged to adjust their means; dragging the shorter red lines changes the common standard deviation.
The bottom of the diagram shows the distribution of X1 + X2. Note that its mean is µ1 + µ2 and its standard deviation is √2 = 1.414 times that of X1 and X2. Change the means and standard deviations of X1 and X2 (by dragging the vertical red lines) and verify that this result holds whatever their distributions.
Click Accumulate then click Take sample a few times to select pairs of random values from the two distributions. Observe that the distributions conform reasonably to the theoretical distributions.
Select Difference from the pop-up menu and repeat with the difference between X1 and X2. Note in particular that:
If µ1 = µ2, then X1 - X2 has a normal distribution with mean zero.