<Dl> <Dd> f (x ∣ μ, σ 2) = 1 σ φ (x − μ σ). (\ displaystyle f (x \ mid \ mu, \ sigma ^ (2)) = (\ frac (1) (\ sigma)) \ varphi \ left ((\ frac (x - \ mu) (\ sigma)) \ right).) </Dd> </Dl> <Dd> f (x ∣ μ, σ 2) = 1 σ φ (x − μ σ). (\ displaystyle f (x \ mid \ mu, \ sigma ^ (2)) = (\ frac (1) (\ sigma)) \ varphi \ left ((\ frac (x - \ mu) (\ sigma)) \ right).) </Dd> <P> The probability density must be scaled by 1 / σ (\ displaystyle 1 / \ sigma) so that the integral is still 1 . </P> <P> If Z (\ displaystyle Z) is a standard normal deviate, then X = σ Z + μ (\ displaystyle X = \ sigma Z+ \ mu) will have a normal distribution with expected value μ (\ displaystyle \ \ mu) and standard deviation σ (\ displaystyle \ \ sigma). Conversely, if X (\ displaystyle X) is a normal deviate with parameters μ (\ displaystyle \ mu) and σ 2 (\ displaystyle \ sigma ^ (2)), then Z = (X − μ) / σ (\ displaystyle Z = (X - \ mu) / \ sigma) will have a standard normal distribution . This variate is called the standardized form of X (\ displaystyle X) </P>

How to find second moment of normal distribution