Normal Distribution
Normal Distribution
Definition
A continuous rv X is said to have a normal distribution with
parameter µ and σ (µ and σ 2 ), where −∞ < µ < ∞ and σ > 0, if
the pdf of X is
f (x; µ, σ) = √
1
2
2
e −(x−µ) /(2σ )
2πσ
We use the notation X ∼ N(µ, σ 2 ) to denote that X is rormally
distributed with parameters µ and σ 2 .
Normal Distribution
Definition
A continuous rv X is said to have a normal distribution with
parameter µ and σ (µ and σ 2 ), where −∞ < µ < ∞ and σ > 0, if
the pdf of X is
f (x; µ, σ) = √
1
2
2
e −(x−µ) /(2σ )
2πσ
We use the notation X ∼ N(µ, σ 2 ) to denote that X is rormally
distributed with parameters µ and σ 2 .
Remark:
1. Obviously, f (x) ≥ 0 for
R ∞all x;1 −(y −µ)2 /(2σ2 )
dy = 1.
2. It is guaranteed that −∞ √2πσ e
Normal Distribution
Normal Distribution
Proposition
For X ∼ N(µ, σ 2 ), we have
E (X ) = µ and V (X ) = σ 2
Normal Distribution
Proposition
For X ∼ N(µ, σ 2 ), we have
E (X ) = µ and V (X ) = σ 2
σ=1
σ=2
σ = 0.5
Normal Distribution
Normal Distribution
The cdf of a normal random variable X is
Z x
F (x) = P(X ≤ x) =
f (y ; µ, σ)dy
−∞
Z x
1
2
2
√
e −(y −µ) /(2σ ) dy
=
2πσ
−∞ Z
x−µ
1
2
2
=√
e −(z) /(2σ ) dz
change of variable:z = y − µ
2πσ −∞
Z x−µ
σ
1
z
2
=√
e −(w ) /2 · σdw
change of variable:w =
σ
2πσ −∞
Z x−µ
σ
1
2
√ e −(w ) /2 dw
=
2π
−∞
Normal Distribution
Normal Distribution
Definition
The normal distribution with parameter values µ = 0 and σ = 1 is
called the standard normal distribution. A random variable
having a standard normal distribution is called a standard normal
random variable and will be denoted by Z . The pdf of Z is
1
2
f (z; 0, 1) = √ e −z /2
2π
−∞<z <∞
The graph of f (z; 0, 1) is called
R z the standard normal (or z) curve.
The cdf of Z is P(Z ≤ z) = −∞ f (y ; 0, 1)dy , which we will
denote by Φ(z).
Normal Distribution
Normal Distribution
Shaded area = Φ(0.5)