Q. What is normal distribution?

Density Function

The normal distribution with location parameter μ and scale parameter σ has probability density function f given by

f(x)=1σϕ(x−μσ)=12π−−−√σexp[−12(x−μσ)2],x∈R

Properties

• f is symmetric about x=μ.
• f $$ increases and then decreases with mode x=μ.
• f is concave upward then downward then upward again, with inflection points at x=μ±σ.
• f (x)→0 as x→∞and as x→−∞

Normal Distribution Function

The normal distribution function F and quantile function F^-1 satisfy the following properties:

• F(x)=Φ(x−μσ)for x∈R.
• F−1(p)=μ+σΦ−1(p) for p∈(0,1).
• F(μ)=12so the median occurs at x=μ.

Moments

If X has the normal distribution with location parameter μ and scale parameter σ, then

• E(x) =  μ
• var(X) = σ²