Download Models of Continuous Random Variables

Models of Discrete Random Variables
Models of Continuous Random Variables
Jiřı́ Neubauer
Department of Econometrics FVL UO Brno
office 69a, tel. 973 442029
email:[email protected]
Jiřı́ Neubauer
Models of Continuous Random Variables
Models of Discrete Random Variables
The
The
The
The
The
Uniform Distribution
Exponential distribution
Normal Distribution
Log-normal Distribution
Pearson, the Student and the Fisher-Snedecor Distribution
The Uniform Distribution
Definition
If the probability density function of X is

1

α < x < β,
f (x) =
β−α

0
otherwise,
where α, β ∈ R, α < β, then X is said to have a uniform distribution on
(α, β), written X ∼ R(α, β)
Jiřı́ Neubauer
Models of Continuous Random Variables
Models of Discrete Random Variables
The
The
The
The
The
Uniform Distribution
Exponential distribution
Normal Distribution
Log-normal Distribution
Pearson, the Student and the Fisher-Snedecor Distribution
The Uniform Distribution
The cumulative distribution function can be calculated as
Zx
F (x) =
Zx
f (t)dt =
−∞
1
x −α
dt = · · · =
β−α
β−α
for
α < x < β.
α
Jiřı́ Neubauer
Models of Continuous Random Variables
Models of Discrete Random Variables
The
The
The
The
The
Uniform Distribution
Exponential distribution
Normal Distribution
Log-normal Distribution
Pearson, the Student and the Fisher-Snedecor Distribution
The Uniform Distribution
The cumulative distribution function can be calculated as
Zx
F (x) =
Zx
f (t)dt =
−∞
1
x −α
dt = · · · =
β−α
β−α
for
α < x < β.
α
We obtain

0

 x −α
F (x) =

 β−α
1
Jiřı́ Neubauer
x ≤ α,
α < x < β,
x ≥ β.
Models of Continuous Random Variables
Models of Discrete Random Variables
The
The
The
The
The
Uniform Distribution
Exponential distribution
Normal Distribution
Log-normal Distribution
Pearson, the Student and the Fisher-Snedecor Distribution
The Uniform Distribution
Figure: The probability density and the cumulative distribution function
R(α, β)
Jiřı́ Neubauer
Models of Continuous Random Variables
Models of Discrete Random Variables
The
The
The
The
The
Uniform Distribution
Exponential distribution
Normal Distribution
Log-normal Distribution
Pearson, the Student and the Fisher-Snedecor Distribution
The Uniform Distribution
The table summarizes some basic information about the uniform
distribution.
E (X )
α+β
2
D(X )
1
12 (β
− α)2
α3 (X )
α4 (X )
quantiles xP
Me(X )
0
−1.2
α + P(β − α)
α+β
2
Jiřı́ Neubauer
Models of Continuous Random Variables
Models of Discrete Random Variables
The
The
The
The
The
Uniform Distribution
Exponential distribution
Normal Distribution
Log-normal Distribution
Pearson, the Student and the Fisher-Snedecor Distribution
The Uniform Distribution
The table summarizes some basic information about the uniform
distribution.
E (X )
α+β
2
D(X )
1
12 (β
− α)2
α3 (X )
α4 (X )
quantiles xP
Me(X )
0
−1.2
α + P(β − α)
α+β
2
Examples: a time we wait for a bus (buses go regularly every 10
minutes), a time we wait for a supply of bread in a grocery store (supplies
are regular), calculation rounding mistakes, . . .
Jiřı́ Neubauer
Models of Continuous Random Variables
Models of Discrete Random Variables
The
The
The
The
The
Uniform Distribution
Exponential distribution
Normal Distribution
Log-normal Distribution
Pearson, the Student and the Fisher-Snedecor Distribution
The Uniform Distribution
Using:
P(X ≤ x0 ) = F (x0 ) =
x0 −α
β−α
for x0 ∈ (α, β)
Jiřı́ Neubauer
Models of Continuous Random Variables
Models of Discrete Random Variables
The
The
The
The
The
Uniform Distribution
Exponential distribution
Normal Distribution
Log-normal Distribution
Pearson, the Student and the Fisher-Snedecor Distribution
The Uniform Distribution
Using:
P(X ≤ x0 ) = F (x0 ) =
x0 −α
β−α
for x0 ∈ (α, β)
P(x1 ≤ X ≤ x2 ) = F (x2 ) − F (x1 ) =
Jiřı́ Neubauer
x2 −α
β−α
−
x1 −α
β−α
for x1 , x2 ∈ (α, β)
Models of Continuous Random Variables
Models of Discrete Random Variables
The
The
The
The
The
Uniform Distribution
Exponential distribution
Normal Distribution
Log-normal Distribution
Pearson, the Student and the Fisher-Snedecor Distribution
Example
Trams go regularly every 10 minutes. The passenger comes to the
tram-stop at the arbitrary time. The random variable X is the time
he/she has to wait for a tram.
Find the probability density function and the distribution function
of X .
Jiřı́ Neubauer
Models of Continuous Random Variables
Models of Discrete Random Variables
The
The
The
The
The
Uniform Distribution
Exponential distribution
Normal Distribution
Log-normal Distribution
Pearson, the Student and the Fisher-Snedecor Distribution
Example
Trams go regularly every 10 minutes. The passenger comes to the
tram-stop at the arbitrary time. The random variable X is the time
he/she has to wait for a tram.
Find the probability density function and the distribution function
of X .
What is the probability that the passenger will wait at most
3 minutes, at least 5 minutes, exactly 7 minutes.
Jiřı́ Neubauer
Models of Continuous Random Variables
Models of Discrete Random Variables
The
The
The
The
The
Uniform Distribution
Exponential distribution
Normal Distribution
Log-normal Distribution
Pearson, the Student and the Fisher-Snedecor Distribution
Example
Trams go regularly every 10 minutes. The passenger comes to the
tram-stop at the arbitrary time. The random variable X is the time
he/she has to wait for a tram.
Find the probability density function and the distribution function
of X .
What is the probability that the passenger will wait at most
3 minutes, at least 5 minutes, exactly 7 minutes.
Calculate the mean, the median, the variance, the standard
deviation and the 90 % quantile.
Jiřı́ Neubauer
Models of Continuous Random Variables
Models of Discrete Random Variables
The
The
The
The
The
Uniform Distribution
Exponential distribution
Normal Distribution
Log-normal Distribution
Pearson, the Student and the Fisher-Snedecor Distribution
Example
The random variable we can describe by the uniform distribution
X ∼ R(0, 10).
Jiřı́ Neubauer
Models of Continuous Random Variables
Models of Discrete Random Variables
The
The
The
The
The
Uniform Distribution
Exponential distribution
Normal Distribution
Log-normal Distribution
Pearson, the Student and the Fisher-Snedecor Distribution
Example
The random variable we can describe by the uniform distribution
X ∼ R(0, 10).
The probability density function is
1
0 < x < 10,
10
f (x) =
0 otherwise,
Jiřı́ Neubauer
Models of Continuous Random Variables
The
The
The
The
The
Models of Discrete Random Variables
Uniform Distribution
Exponential distribution
Normal Distribution
Log-normal Distribution
Pearson, the Student and the Fisher-Snedecor Distribution
Example
The random variable we can describe by the uniform distribution
X ∼ R(0, 10).
The probability density function is
1
0 < x < 10,
10
f (x) =
0 otherwise,
the cumulative distribution function

 0
x
F (x) =
 10
1
Jiřı́ Neubauer
is
x ≤ 0,
0 < x < 10,
x ≥ 10.
Models of Continuous Random Variables
Models of Discrete Random Variables
The
The
The
The
The
Uniform Distribution
Exponential distribution
Normal Distribution
Log-normal Distribution
Pearson, the Student and the Fisher-Snedecor Distribution
Example
Figure: The probability density and the cumulative distribution function
R(0, 10)
Jiřı́ Neubauer
Models of Continuous Random Variables
Models of Discrete Random Variables
The
The
The
The
The
Uniform Distribution
Exponential distribution
Normal Distribution
Log-normal Distribution
Pearson, the Student and the Fisher-Snedecor Distribution
Example
The probability that the passenger will wait
at most 3 minutes
R3 1
P(X ≤ 3) = 10
dx =
0
1
3
10 [x]0
= 0.3
using the distribution function
3
P(X ≤ 3) = F (3) = 10
= 0.3
Jiřı́ Neubauer
Models of Continuous Random Variables
Models of Discrete Random Variables
The
The
The
The
The
Uniform Distribution
Exponential distribution
Normal Distribution
Log-normal Distribution
Pearson, the Student and the Fisher-Snedecor Distribution
Example
The probability that the passenger will wait
at most 3 minutes
R3 1
P(X ≤ 3) = 10
dx =
1
3
10 [x]0
0
= 0.3
using the distribution function
3
P(X ≤ 3) = F (3) = 10
= 0.3
at least 5 minutes
R10 1
P(X ≥ 5) = 10
dx =
5
1
10
10 [x]5
= 0.5
P(X ≥ 5) = 1 − P(X < 5) = 1 − P(X ≤ 5) = 1 − F (5) =
5
= 1 − 10
= 0.5
Jiřı́ Neubauer
Models of Continuous Random Variables
Models of Discrete Random Variables
The
The
The
The
The
Uniform Distribution
Exponential distribution
Normal Distribution
Log-normal Distribution
Pearson, the Student and the Fisher-Snedecor Distribution
Example
The probability that the passenger will wait
at most 3 minutes
R3 1
P(X ≤ 3) = 10
dx =
1
3
10 [x]0
0
= 0.3
using the distribution function
3
P(X ≤ 3) = F (3) = 10
= 0.3
at least 5 minutes
R10 1
P(X ≥ 5) = 10
dx =
5
1
10
10 [x]5
= 0.5
P(X ≥ 5) = 1 − P(X < 5) = 1 − P(X ≤ 5) = 1 − F (5) =
5
= 1 − 10
= 0.5
exactly 7 minutes
P(X = 7) = 0
Jiřı́ Neubauer
Models of Continuous Random Variables
Models of Discrete Random Variables
The
The
The
The
The
Uniform Distribution
Exponential distribution
Normal Distribution
Log-normal Distribution
Pearson, the Student and the Fisher-Snedecor Distribution
Example
the mean E (X ) =
α+β
2
=
10
2
=5
Jiřı́ Neubauer
Models of Continuous Random Variables
Models of Discrete Random Variables
The
The
The
The
The
Uniform Distribution
Exponential distribution
Normal Distribution
Log-normal Distribution
Pearson, the Student and the Fisher-Snedecor Distribution
Example
α+β
10
2 = 2 =5
10
Me(X ) = α+β
2 = 2 =
the mean E (X ) =
the median
Jiřı́ Neubauer
5
Models of Continuous Random Variables
Models of Discrete Random Variables
The
The
The
The
The
Uniform Distribution
Exponential distribution
Normal Distribution
Log-normal Distribution
Pearson, the Student and the Fisher-Snedecor Distribution
Example
α+β
10
2 = 2 =5
10
median Me(X ) = α+β
2 = 2 =5
1
1
variance D(X ) = 12
(β − α)2 = 12
(10
the mean E (X ) =
the
the
Jiřı́ Neubauer
− 0)2 =
100
12
= 8.333
Models of Continuous Random Variables
Models of Discrete Random Variables
The
The
The
The
The
Uniform Distribution
Exponential distribution
Normal Distribution
Log-normal Distribution
Pearson, the Student and the Fisher-Snedecor Distribution
Example
α+β
10
2 = 2 =5
10
median Me(X ) = α+β
2 = 2 =5
1
1
variance D(X ) = 12
(β − α)2 = 12
(10
the mean E (X ) =
the
− 0)2 = 100
12 = 8.333
q
p
the standard variation σ = D(X ) = 100
12 = 2.887
the
Jiřı́ Neubauer
Models of Continuous Random Variables
Models of Discrete Random Variables
The
The
The
The
The
Uniform Distribution
Exponential distribution
Normal Distribution
Log-normal Distribution
Pearson, the Student and the Fisher-Snedecor Distribution
Example
α+β
10
2 = 2 =5
10
median Me(X ) = α+β
2 = 2 =5
1
1
variance D(X ) = 12
(β − α)2 = 12
(10
the mean E (X ) =
the
− 0)2 = 100
12 = 8.333
q
p
the standard variation σ = D(X ) = 100
12 = 2.887
the
90 % quantile x0.90 = α + 0.90(β − α) = 0.9 · 10 = 9
Jiřı́ Neubauer
Models of Continuous Random Variables
Models of Discrete Random Variables
The
The
The
The
The
Uniform Distribution
Exponential distribution
Normal Distribution
Log-normal Distribution
Pearson, the Student and the Fisher-Snedecor Distribution
The Exponential distribution
Definition
If the probability density function of X is
(
1 − x−α
e δ
f (x) =
δ
0
x > α,
x ≤ α,
where α ∈ R, δ > 0, then X is said to have an exponential distribution
with parameters α and δ, written X ∼ Ex(α, δ).
Jiřı́ Neubauer
Models of Continuous Random Variables
Models of Discrete Random Variables
The
The
The
The
The
Uniform Distribution
Exponential distribution
Normal Distribution
Log-normal Distribution
Pearson, the Student and the Fisher-Snedecor Distribution
The Exponential distribution
The cumulative distribution function is
x−α
1 − e− δ
F (x) =
0
Jiřı́ Neubauer
x > α,
x ≤ α.
Models of Continuous Random Variables
Models of Discrete Random Variables
The
The
The
The
The
Uniform Distribution
Exponential distribution
Normal Distribution
Log-normal Distribution
Pearson, the Student and the Fisher-Snedecor Distribution
The Exponential distribution
Figure: The probability density and the cumulative distribution function
Ex(α, δ)
Jiřı́ Neubauer
Models of Continuous Random Variables
Models of Discrete Random Variables
The
The
The
The
The
Uniform Distribution
Exponential distribution
Normal Distribution
Log-normal Distribution
Pearson, the Student and the Fisher-Snedecor Distribution
The Exponential distribution
The table summarizes some basic information about the exponential
distribution.
E (X )
D(X )
α3 (X )
α4 (X )
quantiles xP
Me(X )
α+δ
δ2
2
6
α − δ ln(1 − P)
α + δ ln 2
Jiřı́ Neubauer
Models of Continuous Random Variables
Models of Discrete Random Variables
The
The
The
The
The
Uniform Distribution
Exponential distribution
Normal Distribution
Log-normal Distribution
Pearson, the Student and the Fisher-Snedecor Distribution
The Exponential distribution
The table summarizes some basic information about the exponential
distribution.
E (X )
D(X )
α3 (X )
α4 (X )
quantiles xP
Me(X )
α+δ
δ2
2
6
α − δ ln(1 − P)
α + δ ln 2
Examples: the queuing theory, the reliability theory, the renewal theory,
a time we wait for service, a product lifetime, . . .
Jiřı́ Neubauer
Models of Continuous Random Variables
The
The
The
The
The
Models of Discrete Random Variables
Uniform Distribution
Exponential distribution
Normal Distribution
Log-normal Distribution
Pearson, the Student and the Fisher-Snedecor Distribution
The Exponential distribution
Using:
P(X ≤ x0 ) = F (x0 ) = 1 − e −
Jiřı́ Neubauer
x0 −α
δ
for x0 > α
Models of Continuous Random Variables
The
The
The
The
The
Models of Discrete Random Variables
Uniform Distribution
Exponential distribution
Normal Distribution
Log-normal Distribution
Pearson, the Student and the Fisher-Snedecor Distribution
The Exponential distribution
Using:
P(X ≤ x0 ) = F (x0 ) = 1 − e −
x0 −α
δ
for x0 > α
P(x1 ≤ X ≤ x2 ) = F (x2 ) − F (x1 ) = e −
Jiřı́ Neubauer
x1 −α
δ
− e−
x2 −α
δ
for x1 , x2 > α
Models of Continuous Random Variables
Models of Discrete Random Variables
The
The
The
The
The
Uniform Distribution
Exponential distribution
Normal Distribution
Log-normal Distribution
Pearson, the Student and the Fisher-Snedecor Distribution
Example
It has been found out the time we have to wait for a waiter is a random
variable which has a exponential distribution with the mean 5 minutes
and the standard deviation 2 minutes. Plot the probability density
function and the distribution function. What is the probability that we
will wait at most 5 minutes?
Jiřı́ Neubauer
Models of Continuous Random Variables
Models of Discrete Random Variables
The
The
The
The
The
Uniform Distribution
Exponential distribution
Normal Distribution
Log-normal Distribution
Pearson, the Student and the Fisher-Snedecor Distribution
Example
The mean a the variance of the exponential distribution are
E (X ) = α + δ and D(X ) = δ 2 , thus
α+δ=5
⇒ α = 3, δ = 2
δ=2
Jiřı́ Neubauer
Models of Continuous Random Variables
Models of Discrete Random Variables
The
The
The
The
The
Uniform Distribution
Exponential distribution
Normal Distribution
Log-normal Distribution
Pearson, the Student and the Fisher-Snedecor Distribution
Example
The mean a the variance of the exponential distribution are
E (X ) = α + δ and D(X ) = δ 2 , thus
α+δ=5
⇒ α = 3, δ = 2
δ=2
X ∼ Ex(3, 2)
Jiřı́ Neubauer
Models of Continuous Random Variables
Models of Discrete Random Variables
The
The
The
The
The
Uniform Distribution
Exponential distribution
Normal Distribution
Log-normal Distribution
Pearson, the Student and the Fisher-Snedecor Distribution
Example
Figure: The probability density and the cumulative distribution function
Ex(α, δ)
Jiřı́ Neubauer
Models of Continuous Random Variables
Models of Discrete Random Variables
The
The
The
The
The
Uniform Distribution
Exponential distribution
Normal Distribution
Log-normal Distribution
Pearson, the Student and the Fisher-Snedecor Distribution
Example
The probability that we will wait at most 5 minutes is
P(X ≤ 5) = F (5) = 1 − e −
Jiřı́ Neubauer
5−3
2
= 1 − e −1 = 0.632.
Models of Continuous Random Variables
Models of Discrete Random Variables
The
The
The
The
The
Uniform Distribution
Exponential distribution
Normal Distribution
Log-normal Distribution
Pearson, the Student and the Fisher-Snedecor Distribution
The Normal Distribution
Definition
If X has the probability density function
f (x) =
(x−µ)2
1
√ e − 2σ2
σ 2π
for
x ∈R
where µ ∈ R, σ 2 > 0, it is said to have a normal distribution with
parameters µ and σ 2 , written X ∼ N(µ, σ 2 ).
Jiřı́ Neubauer
Models of Continuous Random Variables
Models of Discrete Random Variables
The
The
The
The
The
Uniform Distribution
Exponential distribution
Normal Distribution
Log-normal Distribution
Pearson, the Student and the Fisher-Snedecor Distribution
The Normal Distribution
The cumulative distribution function is
Zx
F (x) =
f (t)dt =
−∞
1
√
σ 2π
Jiřı́ Neubauer
Zx
e−
(t−µ)2
2σ 2
dt
for
x ∈R
−∞
Models of Continuous Random Variables
Models of Discrete Random Variables
The
The
The
The
The
Uniform Distribution
Exponential distribution
Normal Distribution
Log-normal Distribution
Pearson, the Student and the Fisher-Snedecor Distribution
The Normal Distribution
Figure: The probability density and the cumulative distribution function
N(µ, σ 2 )
Jiřı́ Neubauer
Models of Continuous Random Variables
Models of Discrete Random Variables
The
The
The
The
The
Uniform Distribution
Exponential distribution
Normal Distribution
Log-normal Distribution
Pearson, the Student and the Fisher-Snedecor Distribution
The Normal Distribution
The table summarizes some basic information about the normal
distribution.
E (X )
D(X )
α3 (X )
α4 (X )
quantiles xP
Me(X )
Mo(X )
µ
σ2
0
0
µ + σuP 1
µ
µ
1 quantile
of the standard normal distribution N(0, 1)
Jiřı́ Neubauer
Models of Continuous Random Variables
Models of Discrete Random Variables
The
The
The
The
The
Uniform Distribution
Exponential distribution
Normal Distribution
Log-normal Distribution
Pearson, the Student and the Fisher-Snedecor Distribution
The Normal Distribution
The normally distributed random variable fulfils :
P(µ − σ < X < µ + σ) = 0.683
Jiřı́ Neubauer
Models of Continuous Random Variables
Models of Discrete Random Variables
The
The
The
The
The
Uniform Distribution
Exponential distribution
Normal Distribution
Log-normal Distribution
Pearson, the Student and the Fisher-Snedecor Distribution
The Normal Distribution
The normally distributed random variable fulfils :
P(µ − σ < X < µ + σ) = 0.683
P(µ − 2σ < X < µ + 2σ) = 0.954
Jiřı́ Neubauer
Models of Continuous Random Variables
Models of Discrete Random Variables
The
The
The
The
The
Uniform Distribution
Exponential distribution
Normal Distribution
Log-normal Distribution
Pearson, the Student and the Fisher-Snedecor Distribution
The Normal Distribution
The normally distributed random variable fulfils :
P(µ − σ < X < µ + σ) = 0.683
P(µ − 2σ < X < µ + 2σ) = 0.954
P(µ − 3σ < X < µ + 3σ) = 0.997
Jiřı́ Neubauer
Models of Continuous Random Variables
Models of Discrete Random Variables
The
The
The
The
The
Uniform Distribution
Exponential distribution
Normal Distribution
Log-normal Distribution
Pearson, the Student and the Fisher-Snedecor Distribution
The Normal Distribution
Let us have the random variable X ∼ N(µ, σ 2 ). The transformed random
variable U
X −µ
U=
σ
has the normal distribution with the mean 0 and the variance 1 (the
standard normal distribution U ∼ N(0, 1)).
Jiřı́ Neubauer
Models of Continuous Random Variables
The
The
The
The
The
Models of Discrete Random Variables
Uniform Distribution
Exponential distribution
Normal Distribution
Log-normal Distribution
Pearson, the Student and the Fisher-Snedecor Distribution
The Normal Distribution
The probability density function is
u2
1
φ(u) = √ e − 2
2π
for
u ∈ R,
the cumulative distribution function is
Zu
Φ(u) =
−∞
1
φ(t)dt = √
2π
Jiřı́ Neubauer
Zu
t2
e − 2 dt
for
u∈R
−∞
Models of Continuous Random Variables
Models of Discrete Random Variables
The
The
The
The
The
Uniform Distribution
Exponential distribution
Normal Distribution
Log-normal Distribution
Pearson, the Student and the Fisher-Snedecor Distribution
The Normal Distribution
Figure: The probability density and the cumulative distribution function N(0, 1)
Jiřı́ Neubauer
Models of Continuous Random Variables
Models of Discrete Random Variables
The
The
The
The
The
Uniform Distribution
Exponential distribution
Normal Distribution
Log-normal Distribution
Pearson, the Student and the Fisher-Snedecor Distribution
The Normal Distribution
The table summarizes some basic information about the standard normal
distribution.
E (X )
D(X )
α3 (X )
α4 (X )
kvantily xP
Me(X )
Mo(X )
0
1
0
0
uP 1
0
0
1 the
values are tabulated, for P < 0.5 is uP = −u1−P
Jiřı́ Neubauer
Models of Continuous Random Variables
Models of Discrete Random Variables
The
The
The
The
The
Uniform Distribution
Exponential distribution
Normal Distribution
Log-normal Distribution
Pearson, the Student and the Fisher-Snedecor Distribution
The Normal Distribution
The values of the cumulative distribution function for positive values are
tabulated, for negative values we can write
Φ(−u) = 1 − Φ(u).
Jiřı́ Neubauer
Models of Continuous Random Variables
Models of Discrete Random Variables
The
The
The
The
The
Uniform Distribution
Exponential distribution
Normal Distribution
Log-normal Distribution
Pearson, the Student and the Fisher-Snedecor Distribution
The Normal Distribution
The values of the cumulative distribution function for positive values are
tabulated, for negative values we can write
Φ(−u) = 1 − Φ(u).
If X ∼ N(µ, σ 2 ), U ∼ N(0, 1), then the cumulative distribution function
of the random variable X we can obtain using cumulative distribution
function of U.
x0 −µ
F (x0 ) = P(X ≤ x0 ) = P(X − µ ≤ x0 − µ) = P X −µ
≤
=
σ
σ
x0 −µ
x0 −µ
=P U ≤ σ
=Φ σ
Jiřı́ Neubauer
Models of Continuous Random Variables
Models of Discrete Random Variables
The
The
The
The
The
Uniform Distribution
Exponential distribution
Normal Distribution
Log-normal Distribution
Pearson, the Student and the Fisher-Snedecor Distribution
The Normal Distribution
Quantiles of X (quantiles of U are tabulated):
xP − µ
= Φ(uP ),
F (xP ) = Φ
σ
thus
uP =
xP − µ
⇒ xP = µ + σuP .
σ
Jiřı́ Neubauer
Models of Continuous Random Variables
The
The
The
The
The
Models of Discrete Random Variables
Uniform Distribution
Exponential distribution
Normal Distribution
Log-normal Distribution
Pearson, the Student and the Fisher-Snedecor Distribution
The Normal Distribution
Using:
P(X ≤ x0 ) = F (x0 ) = Φ
x0 −µ
σ
Jiřı́ Neubauer
Models of Continuous Random Variables
The
The
The
The
The
Models of Discrete Random Variables
Uniform Distribution
Exponential distribution
Normal Distribution
Log-normal Distribution
Pearson, the Student and the Fisher-Snedecor Distribution
The Normal Distribution
Using:
P(X ≤ x0 ) = F (x0 ) = Φ
x0 −µ
σ
P(x1 ≤ X ≤ x2 ) = F (x2 ) − F (x1 ) = Φ
Jiřı́ Neubauer
x2 −µ
σ
−Φ
x1 −µ
σ
Models of Continuous Random Variables
Models of Discrete Random Variables
The
The
The
The
The
Uniform Distribution
Exponential distribution
Normal Distribution
Log-normal Distribution
Pearson, the Student and the Fisher-Snedecor Distribution
Example
During quality control we say that the component is acceptable if it’s size
is within the limits 26–27 mm. The size of the component has a normal
distribution with the mean µ = 26.4 mm and the standard deviation
σ = 0.2 mm. What is the probability that the size of the component is
within the given limits?
Jiřı́ Neubauer
Models of Continuous Random Variables
Models of Discrete Random Variables
The
The
The
The
The
Uniform Distribution
Exponential distribution
Normal Distribution
Log-normal Distribution
Pearson, the Student and the Fisher-Snedecor Distribution
Example
The random variable X ∼ N(26.4; 0.22 ).
− Φ 26−26.4
=
P(26 ≤ X ≤ 27) = F (27) − F (26) = Φ 27−26.4
0.2
0.2
= Φ(3) − Φ(−2) = Φ(3) − (1 − Φ(2)) =
= 0.99865 − (1 − 0.97725) = 0.9759
Jiřı́ Neubauer
Models of Continuous Random Variables
Models of Discrete Random Variables
The
The
The
The
The
Uniform Distribution
Exponential distribution
Normal Distribution
Log-normal Distribution
Pearson, the Student and the Fisher-Snedecor Distribution
Example
Figure: The probability density and the cumulative distribution function
N(26.4; 0.04)
Jiřı́ Neubauer
Models of Continuous Random Variables
Models of Discrete Random Variables
The
The
The
The
The
Uniform Distribution
Exponential distribution
Normal Distribution
Log-normal Distribution
Pearson, the Student and the Fisher-Snedecor Distribution
The Log-normal Distribution
Let us assume that X is a non-negative random variable. If a random
variable ln X has a normal distribution N(µ, σ 2 ), then X has a log-normal
distribution LN(µ, σ 2 ).
Definition
If the probability density function of X

(ln x−µ)2
1

√ e − 2σ2
f (x) =
 xσ 2π 0
x > 0,
x ≤ 0,
where µ ≥ 0, σ > 0, then X is said to have a log-normal distribution
with parameters µ and σ 2 , written X ∼ LN(µ, σ 2 ).
Jiřı́ Neubauer
Models of Continuous Random Variables
The
The
The
The
The
Models of Discrete Random Variables
Uniform Distribution
Exponential distribution
Normal Distribution
Log-normal Distribution
Pearson, the Student and the Fisher-Snedecor Distribution
The Log-normal Distribution
The table summarizes some basic information about the log-normal
distribution.
E (X )
D(X )
α3 (X )
α4 (X )
quantiles xP
2
e µ+σ /2
e 2µ ω(ω−1)
√
ω−1(ω+2)
ω 4 +2ω 3 +3ω 2 −6
e µ+σuP
where ω = e σ
2
Jiřı́ Neubauer
Models of Continuous Random Variables
Mo(X )
e µ−σ
2
The
The
The
The
The
Models of Discrete Random Variables
Uniform Distribution
Exponential distribution
Normal Distribution
Log-normal Distribution
Pearson, the Student and the Fisher-Snedecor Distribution
The Log-normal Distribution
The table summarizes some basic information about the log-normal
distribution.
E (X )
D(X )
α3 (X )
α4 (X )
quantiles xP
2
e µ+σ /2
e 2µ ω(ω−1)
√
ω−1(ω+2)
ω 4 +2ω 3 +3ω 2 −6
e µ+σuP
2
Mo(X )
e µ−σ
2
where ω = e σ
Examples: model of entry and wages distributions, a time of renewals,
repairs, the theory of non-coherent particles, . . .
Jiřı́ Neubauer
Models of Continuous Random Variables
The
The
The
The
The
Models of Discrete Random Variables
Uniform Distribution
Exponential distribution
Normal Distribution
Log-normal Distribution
Pearson, the Student and the Fisher-Snedecor Distribution
The Log-normal Distribution
If the random variable X has the log-normal distribution X ∼ LN(µ, σ),
then the transformed random variable
U=
ln X − µ
σ
has the standard normal distribution U ∼ N(0, 1).
Jiřı́ Neubauer
Models of Continuous Random Variables
The
The
The
The
The
Models of Discrete Random Variables
Uniform Distribution
Exponential distribution
Normal Distribution
Log-normal Distribution
Pearson, the Student and the Fisher-Snedecor Distribution
The Log-normal Distribution
If the random variable X has the log-normal distribution X ∼ LN(µ, σ),
then the transformed random variable
U=
ln X − µ
σ
has the standard normal distribution U ∼ N(0, 1).
We can write
ln x0 − µ
F (x0 ) = Φ
= Φ(u),
σ
where Φ(u) is the cumulative distribution function N(0, 1).
Jiřı́ Neubauer
Models of Continuous Random Variables
Models of Discrete Random Variables
The
The
The
The
The
Uniform Distribution
Exponential distribution
Normal Distribution
Log-normal Distribution
Pearson, the Student and the Fisher-Snedecor Distribution
The Log-normal Distribution
Using:
P(X ≤ x0 ) = F (x0 ) = Φ
ln x0 −µ
σ
Jiřı́ Neubauer
Models of Continuous Random Variables
Models of Discrete Random Variables
The
The
The
The
The
Uniform Distribution
Exponential distribution
Normal Distribution
Log-normal Distribution
Pearson, the Student and the Fisher-Snedecor Distribution
The Log-normal Distribution
Using:
P(X ≤ x0 ) = F (x0 ) = Φ
ln x0 −µ
σ
P(x1 ≤ X ≤ x2 ) = F (x2 ) − F (x1 ) = Φ
Jiřı́ Neubauer
ln x2 −µ
σ
−Φ
ln x1 −µ
σ
Models of Continuous Random Variables
Models of Discrete Random Variables
The
The
The
The
The
Uniform Distribution
Exponential distribution
Normal Distribution
Log-normal Distribution
Pearson, the Student and the Fisher-Snedecor Distribution
Example
We suppose that distance between vehicles on the highway (in seconds)
is a random variable which is possible to describe by the log-normal
distribution with parameters µ = 1.27 a σ 2 = 0.49. What is the
probability that the distance will be from 4 till 5 seconds?
Jiřı́ Neubauer
Models of Continuous Random Variables
Models of Discrete Random Variables
The
The
The
The
The
Uniform Distribution
Exponential distribution
Normal Distribution
Log-normal Distribution
Pearson, the Student and the Fisher-Snedecor Distribution
Example
Figure: The probability density and the cumulative distribution function
LN(1.27; 0.7)
Jiřı́ Neubauer
Models of Continuous Random Variables
Models of Discrete Random Variables
The
The
The
The
The
Uniform Distribution
Exponential distribution
Normal Distribution
Log-normal Distribution
Pearson, the Student and the Fisher-Snedecor Distribution
Example
The probability is
P(4 ≤ X ≤ 5) = F (5) − F (4) = Φ ln 5−1.27
−Φ
0.7
= 0.68613 − 0.56597 = 0.12016
Jiřı́ Neubauer
ln 4−1.27
0.7
Models of Continuous Random Variables
=
Models of Discrete Random Variables
The
The
The
The
The
Uniform Distribution
Exponential distribution
Normal Distribution
Log-normal Distribution
Pearson, the Student and the Fisher-Snedecor Distribution
The Pearson χ2 Distribution
Definition
Let us assume that U1 , U2 , . . . , Uν are independent normally distributed
random variables (N(0, 1)). The random variable
χ2 = U12 + U22 + · · · + Uν2 ,
has a χ2 -distribution with ν degrees of freedom.
The parameter ν (the number of freedom) usually represents the number
of independent observation reduced by the number of linear conditions.
Jiřı́ Neubauer
Models of Continuous Random Variables
Models of Discrete Random Variables
The
The
The
The
The
Uniform Distribution
Exponential distribution
Normal Distribution
Log-normal Distribution
Pearson, the Student and the Fisher-Snedecor Distribution
The Pearson χ2 Distribution
Figure: The probability density and the cumulative distribution function χ2 (5)
and χ2 (16)
Jiřı́ Neubauer
Models of Continuous Random Variables
Models of Discrete Random Variables
The
The
The
The
The
Uniform Distribution
Exponential distribution
Normal Distribution
Log-normal Distribution
Pearson, the Student and the Fisher-Snedecor Distribution
The Pearson χ2 Distribution
In the future the quantiles of the χ2 distribution will be useful. They are
usually tabulated for various values P and degrees of freedom ν ≤ 30.
For ν > 30 is possible to use an approximation
χ2P (ν) ≈
2
1 √
2ν − 1 + uP ,
2
where uP is the quantile of N(0, 1).
Jiřı́ Neubauer
Models of Continuous Random Variables
Models of Discrete Random Variables
The
The
The
The
The
Uniform Distribution
Exponential distribution
Normal Distribution
Log-normal Distribution
Pearson, the Student and the Fisher-Snedecor Distribution
The Student distribution t(ν)
Definition
If a random variable U has a standard normal distribution U ∼ N(0, 1), a
random variable χ2 has a Pearson distribution χ2 ∼ χ2 (ν) and if U and
χ2 are independent, then a random variable
U
t=q
χ2
ν
has a Student distribution with ν degrees of freedom, written t ∼ t(ν).
Jiřı́ Neubauer
Models of Continuous Random Variables
Models of Discrete Random Variables
The
The
The
The
The
Uniform Distribution
Exponential distribution
Normal Distribution
Log-normal Distribution
Pearson, the Student and the Fisher-Snedecor Distribution
The Student distribution t(ν)
Figure: The probability density and the cumulative distribution function t(2)
and t(20)
Jiřı́ Neubauer
Models of Continuous Random Variables
Models of Discrete Random Variables
The
The
The
The
The
Uniform Distribution
Exponential distribution
Normal Distribution
Log-normal Distribution
Pearson, the Student and the Fisher-Snedecor Distribution
The Student distribution t(ν)
The probability density function is symmetric with the mean E (t) = 0.
Quantiles of the Student distribution are tabulated for ν ≤ 30 and
P > 0,5, for P < 0, 5 is
tP = −t1−P .
Whether ν > 30, we can use an approximation
tp ≈ uP .
Jiřı́ Neubauer
Models of Continuous Random Variables
Models of Discrete Random Variables
The
The
The
The
The
Uniform Distribution
Exponential distribution
Normal Distribution
Log-normal Distribution
Pearson, the Student and the Fisher-Snedecor Distribution
The Fisher-Snedecor Distribution F (ν1 , ν2 )
Definition
If a random variable χ21 has χ21 ∼ χ2 (ν1 ) with ν1 degrees of freedom and
a random variable χ22 has χ22 ∼ χ2 (ν2 ) with ν2 degrees of freedom and
they are independent, then a random variable
F =
χ21 χ22
:
ν1 ν2
has a Fisher-Snedecor distribution with ν1 and ν2 degrees of freedom,
written F ∼ F (ν1 , ν2 ) .
Jiřı́ Neubauer
Models of Continuous Random Variables
Models of Discrete Random Variables
The
The
The
The
The
Uniform Distribution
Exponential distribution
Normal Distribution
Log-normal Distribution
Pearson, the Student and the Fisher-Snedecor Distribution
The Fisher-Snedecor Distribution F (ν1 , ν2 )
Figure: The probability density and the cumulative distribution function
F (30, 20) and F (3, 50)
Jiřı́ Neubauer
Models of Continuous Random Variables
Models of Discrete Random Variables
The
The
The
The
The
Uniform Distribution
Exponential distribution
Normal Distribution
Log-normal Distribution
Pearson, the Student and the Fisher-Snedecor Distribution
The Fisher-Snedecor Distribution F (ν1 , ν2 )
The Fischer-Snedecor distribution is asymmetric.
Quantiles of F distribution are tabulated for P > 0,5, for P < 0,5 we can
use the formula
1
FP (ν1 , ν2 ) =
.
F1−P (ν2 , ν1 )
Jiřı́ Neubauer
Models of Continuous Random Variables