Download MAS113 Introduction to Probability and Statistics

Normal distribution – revision
MAS113 Introduction to Probability and
Statistics
Dr Jonathan Jordan
School of Mathematics and Statistics, University of Sheffield
Spring Semester, 2017
Dr Jonathan Jordan
MAS113 Introduction to Probability and Statistics
Normal distribution – revision
Definition
Definition
If a random variable X has a normal distribution with mean
µ and variance σ 2 , then its probability density function is given
by
1
1
2
fX (x) = √
exp − 2 (x − µ)
2σ
2πσ 2
Dr Jonathan Jordan
MAS113 Introduction to Probability and Statistics
Normal distribution – revision
Definition
Definition
If a random variable X has a normal distribution with mean
µ and variance σ 2 , then its probability density function is given
by
1
1
2
fX (x) = √
exp − 2 (x − µ)
2σ
2πσ 2
We write
X ∼ N(µ, σ 2 ),
to mean “X has a normal distribution with mean µ and
variance σ 2 .”
Dr Jonathan Jordan
MAS113 Introduction to Probability and Statistics
Normal distribution – revision
Properties
E (X ) = µ and Var(X ) = σ 2
Dr Jonathan Jordan
MAS113 Introduction to Probability and Statistics
Normal distribution – revision
Properties
E (X ) = µ and Var(X ) = σ 2
If Z ∼ N(0, 1) Z has a standard normal distribution.
Dr Jonathan Jordan
MAS113 Introduction to Probability and Statistics
Normal distribution – revision
Properties
E (X ) = µ and Var(X ) = σ 2
If Z ∼ N(0, 1) Z has a standard normal distribution.
Distribution function of standard normal
Z z
t2
1
√ e − 2 dt
Φ(z) = P(Z ≤ z) =
2π
−∞
Dr Jonathan Jordan
MAS113 Introduction to Probability and Statistics
Normal distribution – revision
Properties
E (X ) = µ and Var(X ) = σ 2
If Z ∼ N(0, 1) Z has a standard normal distribution.
Distribution function of standard normal
Z z
t2
1
√ e − 2 dt
Φ(z) = P(Z ≤ z) =
2π
−∞
given by pnorm in R.
Dr Jonathan Jordan
MAS113 Introduction to Probability and Statistics
Normal distribution – revision
Standardisation
If Z ∼ N(0, 1) then µ + σZ ∼ N(µ, σ 2 ).
Dr Jonathan Jordan
MAS113 Introduction to Probability and Statistics
Normal distribution – revision
Standardisation
If Z ∼ N(0, 1) then µ + σZ ∼ N(µ, σ 2 ).
If X ∼ N(µ, σ 2 ) then
X −µ
∼ N(0, 1).
σ
Dr Jonathan Jordan
MAS113 Introduction to Probability and Statistics
Normal distribution – revision
The two-σ rule
For a standard normal random variable Z ,
P(−1.96 ≤ Z ≤ 1.96) = 0.95.
Dr Jonathan Jordan
MAS113 Introduction to Probability and Statistics
Normal distribution – revision
The two-σ rule
For a standard normal random variable Z ,
P(−1.96 ≤ Z ≤ 1.96) = 0.95.
Since E (Z ) = 0 and Var(Z ) = 1, the probability of Z being
within two standard deviations of its mean value is
approximately 0.95 (ie P(−2 ≤ Z ≤ 2) = 0.9545 to 4 d.p.).
Dr Jonathan Jordan
MAS113 Introduction to Probability and Statistics
Normal distribution – revision
The two-σ rule: general case
If we now consider any normal random variable X ∼ N(µ, σ 2 ),
the probability that it will lie within a distance of two standard
deviations from its mean is approximately 0.95.
Dr Jonathan Jordan
MAS113 Introduction to Probability and Statistics
Normal distribution – revision
The two-σ rule: general case
If we now consider any normal random variable X ∼ N(µ, σ 2 ),
the probability that it will lie within a distance of two standard
deviations from its mean is approximately 0.95.
This is straightforward to verify:
Dr Jonathan Jordan
MAS113 Introduction to Probability and Statistics
Normal distribution – revision
The two-σ rule: general case
If we now consider any normal random variable X ∼ N(µ, σ 2 ),
the probability that it will lie within a distance of two standard
deviations from its mean is approximately 0.95.
This is straightforward to verify:
(on board)
Dr Jonathan Jordan
MAS113 Introduction to Probability and Statistics
Normal distribution – revision
Convention
In Statistics, there is a convention of using 0.05 as a threshold
for a ‘small’ probability, though the choice of 0.05 is arbitrary.
Dr Jonathan Jordan
MAS113 Introduction to Probability and Statistics
Normal distribution – revision
Convention
In Statistics, there is a convention of using 0.05 as a threshold
for a ‘small’ probability, though the choice of 0.05 is arbitrary.
However, the two-σ rule is an easy to remember fact about
normal random variables, and can be a useful yardstick in
various situations.
Dr Jonathan Jordan
MAS113 Introduction to Probability and Statistics