Normal distribution addition

If we add together two normal distributions (they can be multiples) and an optional constant, the result will also follow a normal distribution. That’s because the normal distribution is a scalable distribution.

Z=aX+bY+c

• X, Y and Z are independent and random variables following a normal distribution
• a, b and c are constants.

See continuous random expectation transformation and continuous random variance transformation for how to find aX and bY.

Subtraction

We can’t easily subtract two normal distributions, but we can add them if one of them is negative.

If Z=aX-bY:

• E(Z)=E(aX)+E(-bY)=aE(X)-bE(Y)
• Var(Z)=Var(aX)+Var(-bY)=a^2Var(X)+b^2Var(Y)
  • the negatives disappear because we are squaring the -b