Download Continuous Random Variables Chapter 5 Important Concepts

Continuous Random Variables
Chapter 5
Important Concepts
Continuous Random Variables
Introductory Example
Random Angles
• Distribution Functions.
For the Uniform Spinner,
• Density Functions.
Ω = (−π, π],
• The Median and Other Quantiles.
P (a, b] =
• The Mean and Variance
b−a
2π
for −π ≤ a < b ≤ π. Let
• Special Distributions
X(ω) = ω
Uniform
for −π < ω ≤ π. Then X is random variable.
Exponential
Note: P [X = x] = 0 for all x ∈ IR.
Normal
Two Important Relations
Distributions Functions
If X ∼ F , then
Def. Given a model (Ω, P ) and a RV
P [a < X ≤ b] = F (b) − F (a)
X : Ω → IR,
for all −∞ < a < b < ∞, since
the distribution function of X is defined by
F (x) = P [X ≤ x]
(∗)
for −∞ < x < ∞.
= F (a) + P [a < X ≤ b]
Also,
Example. For the random angle
F (x) =
F (b) = P [X ≤ b] = P [X ≤ a] + P [a < X ≤ b]
1
x
+
2 2π
for −π < x ≤ π, since
P [X ≤ x] = P [−π < X ≤ x]
x+π
= P ((π, x]) =
.
2π
Notation: Write X ∼ F if X is a RV with DF F ;
that is, if (*) holds.
P [X > b] = 1 − P [X ≤ b] = 1 − F (b).
Example. In the random angle example,
F (x) =
1
x
+
,
2 2π
2
1
1 1
1 1
F ( π) − F ( π) = ( + ) − ( + )
3
3
2 3
2 6
1
= .
6
Some Definitions
Def. A function F : IR → IR is non-decreasing if
F (x) ≤ F (y) whenver x ≤ y.
Characteristic Properties
One Sided Limits
F (x+) =
lim
F (y),
lim
F (y).
y→x,y>x
and
F (x−) =
y→x,y<x
Theorem. A function F : IR → IR is the
distribution function of some RV iff
F (x) ≤ F (y) whenever x ≤ y,
(1)
F (x) = lim F (y) each all x,
(2)
y↓x
lim F (x) = 0,
(3a)
lim F (x) = 1.
(3b)
x→−∞
x→∞
Proof. Later.
Note: Henceforth DF means a function satisfying
(1), (2), and (3).
The Discrete Case
Densities
If X is discrete with range X and PMF f , then
F (x) = P [X ≤ x]
X
=
f (y).
y∈X ,y≤x
If also
Def. A function f : IR → IR is a density if
f (x) ≥ 0, for all x,
Z ∞
f (x)dx = 1.
−∞
X ⊆ {· · · − 1, 0, 1, 2, · · · },
then
F (n) =
X
f (k)
k≤n
If X ∼ F and f is a density then X and F have
density f if
Z x
f (y)dy
(4)
F (x) =
−∞
for all −∞ < x < ∞.
and
f (n) = F (n) − F (n − 1).
Theorem. If f is any density, then (4) defines a
DF.
Proof. Exercise.
Corollary. If f is any density, then there is a RV
X with density f .
Example
Uniform Distributions
Consequences Of
F (x) =
Z
x
f (y)dy
(4)
−∞
If X ∼ F with density f , then
P [a < X ≤ b] = F (b) − F (a) =
Z
b
f (x)dx
a
for −∞ ≤ a < b ≤ ∞.
If −∞ < α < β < ∞, then

1/(β − α) if α < x ≤ β
f (x) =
0
if otherwise
is a density, since f (x) ≥ 0 for all x and
Z ∞
Z β
dx
f (x)dx =
β
−α
α
∞
1
(β − α) = 1.
=
β−α
Then
If (4) holds, then


if x ≤ α

0
F (x) = (x − α)/(β − α) if α < x ≤ β



1
if x > β
d
F (x)
f (x) = F 0 (x) =
dx
at continuity points of f .
Example: Random Angles: α = −π and β = π.
Standard α = 0 and β = 1.
The Standard Normal Density
Fact
Example
Exponential Distributions
If 0 < λ < ∞, then

0
F (x) =
1 − e−λx
if x ≤ 0
if else
has density
f (x) =
Z
∞
1
2
e− 2 z dz =
√
2π.
−∞
Let
1 2
1
ϕ(z) = √ e− 2 z
2π
Then ϕ is a density, called the standard normal
density. The standard normal distribution
function is
Z z
Φ(z) =
ϕ(y)dy.
−∞

0
λe
−λx
if − ∞ < x < 0
if 0 ≤ x < ∞.
Note: Tabled
Note:
Φ(−z) =
Notes a). Derivative doesn’t exist when x = 0.
=
b). Doesn’t matter.
−z
Z
ϕ(y)dy
−∞
Z ∞
z
ϕ(x)dx = 1 − Φ(z)
and
Φ(0) =
1
.
2
3
2
1
x
0
Standard Normal Density
The Normal Distribution
With Parameters
Let
−2
−1
1
f (x) = ϕ
σ
−3
Z
x
f (y)dy = Φ
−∞
0.0
0.1
0.2
0.3
0.4
Density
F (m) =
th
If
F (t) =
F (x) = p
then

0
1 − e−λt
F (m) =
P [X ≤ x] = p.
x−µ
σ
if ∞ < t < 0
if 0 ≤ t < ∞
1
2
iff
1 − e−λm =
th
is called a p -quantile or 100p percentile of X
or F . In terms of X, the condition is
Example
1
2
is called a median of X or F . More generally, if
0 < p < 1, then any x for which
These are the normal distribution with
parameters −∞ < µ < ∞ and σ > 0.
The Median and Other Quantiles
If X ∼ F , then and m for which
x−µ
σ
and
F (x) =
Figure 1: The Standard Normal Density
iff
e−λm =
1
2
1
2
iff
eλm = 2
iff
1
log(2),
λ
where log denotes natural logarthim.
m=
Example: The half life of a radio active
substance.
The Mean and Variance
Def. If X has density f , then the mean of X
and/or f is defined by
Z ∞
xf (x)dx,
µ=
−∞
Variance And Standard Deviation
provided that the integral converges absolutely.
Example: Uniform Distributions. If

1/(β − α) if α < x ≤ β
f (x) =
0
if otherwise
The variance of X and/or f is
Z ∞
(x − µ)2 f (x)dx.
σ2 =
−∞
Also,
then
σ =
Z
∞
−∞
β
xdx
α β−α
1 x2 β
=
|
2 β − α x=α
1 β 2 − α2
=
2 β−α
α+β
,
=
2
µ=
Z
2
x2 f (x) − µ2
The standard deviation is
√
σ = σ2 .
since β 2 − α2 = (β + α)(β − α).
An Example
A Uniform Distribution
Normal Distribution Function
If

1 if 0 ≤ x ≤ 1
f (x) =
0 if otherwise
then
Z
µ=
Z
1
Z
1
1
xdx =
0
1 21
1
= ,
x |
2 x=0 2
and
σ =
are both µ, since
1
1
x dx = x3 |1x=0 =
3
3
2
0
2
The Median and Mean The mean and median
of a normal distribution function
x−µ
F (x) = Φ
σ
1
,
2
and
1
1 1
.
x dx − µ = − =
3 4
12
2
0
F (µ) = Φ(0) =
2
Z
∞
xf (x)dx = µ +
−∞
Z
∞
(x − µ)f (x)dx
−∞
Z ∞
=µ+σ
More generally, if
zϕ(z)dz = µ.
−∞
X ∼ Unif(α, β),
then
σ2 =
(β − α)2
.
12
The Variance Similarly, the variance is σ 2 .
The Gamma Function
Let
Γ(α) =
for α > 0. Then
Z
Γ(1) =
Z
∞
The Mean and Variance Of
Exponential Distributions
xα−1 e−x dx
0
If
∞
e
−x
dx =
0
−e−x |∞
x=0
= 1,
and
Γ(α + 1) = αΓ(α)
for α > 0. For, integrating by parts,
Z ∞
Γ(α) =
xα e−x dx
0
Z ∞
= −xα e−x |∞
+
αxα−1 e−x dx
x=0
0
= αΓ(α).
So
Γ(n) = (n − 1)!.
The Mean and Variance Of
Exponential Distributions
Continued
Similarly,
Z
∞
x2 λe−λx dx
0
Z ∞2
1
x2 e−λx dx
λ2 0
1
= 2 Γ(3)
λ
2
= 2
λ
=
and
σ2 = 2
λ2
− µ2 =
1
.
λ2
f (x) =
then

0
if x < 0
λe−λx
µ=
Z
if 0 ≤ x < ∞
∞
xλe−λx dx
0
Z
1 ∞ −λx
xe
dx
λ 0
1
= Γ(2)
λ
1
=
λ
=