Download Normal (Gaussian) Distribution

Normal (Gaussian) Distribution
Dennis Sun
Normal Distribution
A Normal(µ, σ 2 ) random variable is one whose p.d.f. is
p(x)
1
2
1
p(x) = √ e− 2σ2 (x−µ)
σ 2π
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0 4
−∞ < x < ∞.
µ = 0, σ 2 = 1
µ = 1, σ 2 = . 25
µ = − 1, σ 2 = 2
3
2
1 0 1 2 3 4
x
Notice that the standard normal is a Normal(0, 1) distribution.
The normal distribution is also called the Gaussian distribution,
after the mathematician Carl Friedrich Gauss.
Location-Scale Transformations
Let Z ∼ Normal(0, 1).
• What is the distribution of X = Z + µ?
X ∼ Normal(µ, 1).
• What is the distribution of X = σZ?
X ∼ Normal(0, σ 2 ).
We can get any X ∼ Normal(µ, σ 2 ) as a location-scale transform
of Z ∼ Normal(0, 1):
X = µ + σZ.
Notice that this implies E[X] = µ and Var[X] = σ 2 .
Standardization
We can write any X ∼ Normal(µ, σ 2 ) as
X = µ + σZ,
where Z ∼ N (0, 1).
This means that we can transform any X ∼ Normal(µ, σ 2 ) into a
standard normal as follows:
Z=
X −µ
.
σ
This process is known as standardization.
Standardization is very useful for calculating probabilities involving
X because we have a table for the c.d.f. Φ of Z.
Example
A voltage signal is sent over a communications channel. The
channel adds Gaussian noise with mean 0 V and standard
deviation 1.25 V to whatever voltage is sent.
That is, if the sender sends voltage v, the receiver sees X = v + N ,
where N ∼ Normal(0, 1.252 ). If a +2 V signal is sent, find the
probability that the receiver sees a negative voltage.