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.

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

Question	Answer
What is the result when adding two normal distributions (with optional constant)?	The result also follows a normal distribution because the normal distribution is scalable.
Given Z = aX + bY + c, what conditions apply to X, Y, and Z?	X, Y, and Z are independent random variables following a normal distribution; a, b, and c are constants.
How can we handle subtraction of two normal distributions like Z = aX - bY?	We treat it as addition with a negative coefficient, using E(Z)=aE(X)-bE(Y) and Var(Z)=a^2Var(X)+b^2Var(Y).
Why does the negative sign disappear in the variance formula for Z = aX - bY?	Because variance calculation squares the coefficient: Var(-bY) = (-b)^2 Var(Y) = b^2 Var(Y).