The family of normal distributions is closed under linear transformations: if X is normally distributed with mean μ and standard deviation σ, then the variable Y = aX + b, for any real numbers a and b, is also normally distributed, with mean aμ + b and standard deviation
Also if X1 and X2 are two independent normal random variables, with means μ1, μ2 and standard deviations σ1, σ2, then their sum X1 + X2 will also be normally distributed,[proof] with mean μ1 + μ2 and variance sigma_1^2 + sigma_2^2.
In particular, if X and Y are independent normal deviates with zero mean and variance σ2, then X + Y and X − Y are also independent and normally distributed, with zero mean and variance 2σ2. This is a special case of the polarization identity.
Also, if X1, X2 are two independent normal deviates with mean μ and deviation σ, and a, b are arbitrary real numbers, then the variable
X_3 = frac{aX_1 + bX_2 - (a+b)mu}{sqrt{a^2+b^2}} + mu
is also normally distributed with mean μ and deviation σ. It follows that the normal distribution is stable (with exponent α = 2).
More generally, any linear combination of independent normal deviates is a normal deviate.