Download Lecture 5: Probabilities and distributions (Part 3)

Some News Continuous Joint PD Continuous Marginal Distributions
Conditional Probabilities Conditional Probability D
Lecture 5: Probabilities and distributions
(Part 3)
Kateřina Staňková
Statistics (MAT1003)
April 19, 2012
beamer-tu-log
Some News Continuous Joint PD Continuous Marginal Distributions
Conditional Probabilities Conditional Probability D
Outline
1
Some News
2
Continuous Joint PD
3
Continuous Marginal Distributions
4
Conditional Probabilities
5
Conditional Probability Distributions
Discrete
Continuous = Density
beamer-tu-log
Some News Continuous Joint PD Continuous Marginal Distributions
Conditional Probabilities Conditional Probability D
And now . . .
1
Some News
2
Continuous Joint PD
3
Continuous Marginal Distributions
4
Conditional Probabilities
5
Conditional Probability Distributions
Discrete
Continuous = Density
beamer-tu-log
Some News Continuous Joint PD Continuous Marginal Distributions
Conditional Probabilities Conditional Probability D
News
Homework deadline: almost one day later: April 24, 13.35;
also new homework will be assigned on Tuesday
Absence: Measured from
2
3
of classes
April 23: Exercise session ⇒ lot of theory today
beamer-tu-log
Some News Continuous Joint PD Continuous Marginal Distributions
Conditional Probabilities Conditional Probability D
News
Homework deadline: almost one day later: April 24, 13.35;
also new homework will be assigned on Tuesday
Absence: Measured from
2
3
of classes
April 23: Exercise session ⇒ lot of theory today
beamer-tu-log
Some News Continuous Joint PD Continuous Marginal Distributions
Conditional Probabilities Conditional Probability D
News
Homework deadline: almost one day later: April 24, 13.35;
also new homework will be assigned on Tuesday
Absence: Measured from
2
3
of classes
April 23: Exercise session ⇒ lot of theory today
beamer-tu-log
Some News Continuous Joint PD Continuous Marginal Distributions
Conditional Probabilities Conditional Probability D
News
Homework deadline: almost one day later: April 24, 13.35;
also new homework will be assigned on Tuesday
Absence: Measured from
2
3
of classes
April 23: Exercise session ⇒ lot of theory today
beamer-tu-log
Some News Continuous Joint PD Continuous Marginal Distributions
Conditional Probabilities Conditional Probability D
And now . . .
1
Some News
2
Continuous Joint PD
3
Continuous Marginal Distributions
4
Conditional Probabilities
5
Conditional Probability Distributions
Discrete
Continuous = Density
beamer-tu-log
Some News Continuous Joint PD Continuous Marginal Distributions
Conditional Probabilities Conditional Probability D
book: Section 3.4
Continuous Joint Probability Distribution
Let X be a continuous RV with probability density function f .
Then
d
P(X ≤ x)
dx
P(X ≤ x + ∆ x) − P(X ≤ x)
= lim
∆x
∆x→0
P(x ≤ X ≤ x + ∆ x)
= lim
∆x
∆x→0
f (x) = F 0 (x) =
Analogously for 2 continuous RV’s X and Y :
f (x, y ) =
lim
(δ,)→(0,0)
P(x ≤ X ≤ x + δ, y ≤ Y ≤ y + )
δ
is the joint probability density function.
beamer-tu-log
Some News Continuous Joint PD Continuous Marginal Distributions
Conditional Probabilities Conditional Probability D
book: Section 3.4
Continuous Joint Probability Distribution
Let X be a continuous RV with probability density function f .
Then
d
P(X ≤ x)
dx
P(X ≤ x + ∆ x) − P(X ≤ x)
= lim
∆x
∆x→0
P(x ≤ X ≤ x + ∆ x)
= lim
∆x
∆x→0
f (x) = F 0 (x) =
Analogously for 2 continuous RV’s X and Y :
f (x, y ) =
lim
(δ,)→(0,0)
P(x ≤ X ≤ x + δ, y ≤ Y ≤ y + )
δ
is the joint probability density function.
beamer-tu-log
Some News Continuous Joint PD Continuous Marginal Distributions
Conditional Probabilities Conditional Probability D
Continuous Joint Probability Distribution
For 2 continuous RV’s X and Y :
f (x, y ) =
P(x ≤ X ≤ x + δ, y ≤ Y ≤ y + )
δ
(δ,)→(0,0)
lim
is the joint probability density function.
Properties of Continuous Joint PD
1
2
f (x, y ) ≥ 0 for all x, y ∈ R
RR
R +∞ R +∞
f (x, y )dxdy = −∞ −∞ f (x, y )dxdy = 1
R2
3
RR
f (x, y )dxdy = P ((x, y ) ∈ A) for all A ⊆ R2
A
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Some News Continuous Joint PD Continuous Marginal Distributions
Conditional Probabilities Conditional Probability D
Continuous Joint Probability Distribution
For 2 continuous RV’s X and Y :
f (x, y ) =
P(x ≤ X ≤ x + δ, y ≤ Y ≤ y + )
δ
(δ,)→(0,0)
lim
is the joint probability density function.
Properties of Continuous Joint PD
1
2
f (x, y ) ≥ 0 for all x, y ∈ R
RR
R +∞ R +∞
f (x, y )dxdy = −∞ −∞ f (x, y )dxdy = 1
R2
3
RR
f (x, y )dxdy = P ((x, y ) ∈ A) for all A ⊆ R2
A
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Some News Continuous Joint PD Continuous Marginal Distributions
Conditional Probabilities Conditional Probability D
Continuous Joint Probability Distribution
For 2 continuous RV’s X and Y :
f (x, y ) =
P(x ≤ X ≤ x + δ, y ≤ Y ≤ y + )
δ
(δ,)→(0,0)
lim
is the joint probability density function.
Properties of Continuous Joint PD
1
2
f (x, y ) ≥ 0 for all x, y ∈ R
RR
R +∞ R +∞
f (x, y )dxdy = −∞ −∞ f (x, y )dxdy = 1
R2
3
RR
f (x, y )dxdy = P ((x, y ) ∈ A) for all A ⊆ R2
A
beamer-tu-log
Some News Continuous Joint PD Continuous Marginal Distributions
Conditional Probabilities Conditional Probability D
Continuous Joint Probability Distribution
For 2 continuous RV’s X and Y :
f (x, y ) =
P(x ≤ X ≤ x + δ, y ≤ Y ≤ y + )
δ
(δ,)→(0,0)
lim
is the joint probability density function.
Properties of Continuous Joint PD
1
2
f (x, y ) ≥ 0 for all x, y ∈ R
RR
R +∞ R +∞
f (x, y )dxdy = −∞ −∞ f (x, y )dxdy = 1
R2
3
RR
f (x, y )dxdy = P ((x, y ) ∈ A) for all A ⊆ R2
A
beamer-tu-log
Some News Continuous Joint PD Continuous Marginal Distributions
Conditional Probabilities Conditional Probability D
Continuous Joint Probability Distribution
For 2 continuous RV’s X and Y :
f (x, y ) =
P(x ≤ X ≤ x + δ, y ≤ Y ≤ y + )
δ
(δ,)→(0,0)
lim
is the joint probability density function.
Properties of Continuous Joint PD
1
2
f (x, y ) ≥ 0 for all x, y ∈ R
RR
R +∞ R +∞
f (x, y )dxdy = −∞ −∞ f (x, y )dxdy = 1
R2
3
RR
f (x, y )dxdy = P ((x, y ) ∈ A) for all A ⊆ R2
A
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Some News Continuous Joint PD Continuous Marginal Distributions
Conditional Probabilities Conditional Probability D
Properties of Continuous Joint PD
1
2
f (x, y ) ≥ 0 for all x, y ∈ R
RR
f (x, y )dxdy = 1
R2
3
RR
f (x, y )dxdy = P ((x, y ) ∈ A) for all A ⊆ R2
A
Example 1
X , Y RV’s with
f (x, y ) =
y
,
2 x2
0,
1
2
< x < 1, 0 < y < 2
elsewhere
(a) Validate condition 2. (b) Calculate P(X ≥ 53 )
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Some News Continuous Joint PD Continuous Marginal Distributions
Conditional Probabilities Conditional Probability D
Properties of Continuous Joint PD
1
2
f (x, y ) ≥ 0 for all x, y ∈ R
RR
f (x, y )dxdy = 1
R2
3
RR
f (x, y )dxdy = P ((x, y ) ∈ A) for all A ⊆ R2
A
Example 1
X , Y RV’s with
f (x, y ) =
y
,
2 x2
0,
1
2
< x < 1, 0 < y < 2
elsewhere
(a) Validate condition 2. (b) Calculate P(X ≥ 53 )
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Some News Continuous Joint PD Continuous Marginal Distributions
Example 1(a): Verify
(f (x, y ) =
y
,
2 x2
if
1
2
RR
Conditional Probabilities Conditional Probability D
f (x, y )dxdy = 1
R2
< x < 1, 0 < y < 2 and 0 elsewhere )
Z Z
1Z 2
Z
f (x, y )dxdy =
1
2
R2
0
1
Z
=
1
2
y
dy dx
2 x2
y2
4 x2
2
1
Z
dx =
0
1
2
1
1
1
d
x
=
−
=1
x 1
x2
2
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Some News Continuous Joint PD Continuous Marginal Distributions
Example 1(a): Verify
(f (x, y ) =
y
,
2 x2
if
1
2
RR
Conditional Probabilities Conditional Probability D
f (x, y )dxdy = 1
R2
< x < 1, 0 < y < 2 and 0 elsewhere )
Z Z
1Z 2
Z
f (x, y )dxdy =
1
2
R2
0
1
Z
=
1
2
y
dy dx
2 x2
y2
4 x2
2
1
Z
dx =
0
1
2
1
1
1
d
x
=
−
=1
x 1
x2
2
Example 1(b): Calculate P(X ≥ 53 )
3
P(X ≥ ) =
5
1Z 2
Z
3
5
0
y
dy dx =
2 x2
1
Z
3
5
1
2
dx =
2
3
x
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Some News Continuous Joint PD Continuous Marginal Distributions
Conditional Probabilities Conditional Probability D
Properties of Continuous Joint PD
1
2
f (x, y ) ≥ 0 for all x, y ∈ R
RR
f (x, y )dxdy = 1
R2
3
RR
f (x, y )dxdy = P ((x, y ) ∈ A) for all A ⊆ R2
A
Example 2
X , Y RV’s with
f (x, y ) =
10 x y 2 , 0 < x < y < 1
0, elsewhere
(a) Validate condition 2. (b) Calculate P( 14 ≤ Y ≤ 12 )
(c) Calculate P( 81 ≤ X ≤ 38 , 41 ≤ Y ≤ 12 )
beamer-tu-log
Some News Continuous Joint PD Continuous Marginal Distributions
Conditional Probabilities Conditional Probability D
Properties of Continuous Joint PD
1
2
f (x, y ) ≥ 0 for all x, y ∈ R
RR
f (x, y )dxdy = 1
R2
3
RR
f (x, y )dxdy = P ((x, y ) ∈ A) for all A ⊆ R2
A
Example 2
X , Y RV’s with
f (x, y ) =
10 x y 2 , 0 < x < y < 1
0, elsewhere
(a) Validate condition 2. (b) Calculate P( 14 ≤ Y ≤ 12 )
(c) Calculate P( 81 ≤ X ≤ 38 , 41 ≤ Y ≤ 12 )
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Some News Continuous Joint PD Continuous Marginal Distributions
Example 2(a): Verify
RR
Conditional Probabilities Conditional Probability D
f (x, y )dxdy = 1
R2
10 x y 2 , 0 < x < y < 1
f (x, y ) =
0, elsewhere
Find the proper area:
Either we notice that x is between 0 and 1 and that y is
between x and 1
Or: y between 0 and 1 and x between 0 and y
beamer-tu-log
Some News Continuous Joint PD Continuous Marginal Distributions
Example 2(a): Verify
RR
Conditional Probabilities Conditional Probability D
f (x, y )dxdy = 1
R2
10 x y 2 , 0 < x < y < 1
f (x, y ) =
0, elsewhere
Find the proper area:
Either we notice that x is between 0 and 1 and that y is
between x and 1
Or: y between 0 and 1 and x between 0 and y
beamer-tu-log
Some News Continuous Joint PD Continuous Marginal Distributions
Example 2(a): Verify
RR
Conditional Probabilities Conditional Probability D
f (x, y )dxdy = 1
R2
10 x y 2 , 0 < x < y < 1
f (x, y ) =
0, elsewhere
Find the proper area:
Either we notice that x is between 0 and 1 and that y is
between x and 1
Or: y between 0 and 1 and x between 0 and y
beamer-tu-log
Some News Continuous Joint PD Continuous Marginal Distributions
Example 2(a): Verify
RR
Conditional Probabilities Conditional Probability D
f (x, y )dxdy = 1
R2
10 x y 2 , 0 < x < y < 1
f (x, y ) =
0, elsewhere
Find the proper area:
Either we notice that x is between 0 and 1 and that y is
between x and 1
Or: y between 0 and 1 and x between 0 and y
Z Z
Z
1Z 1
f (x, y )dxdy =
0
R2
Z
x
1Z
=
0
10 xy 2 dy dx
0
y
10 x y 2 dxdy = 1
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Some News Continuous Joint PD Continuous Marginal Distributions
Conditional Probabilities Conditional Probability D
Example 2(b): Calculate P( 14 ≤ Y ≤ 12 )
10 x y 2 , 0 < x < y < 1
f (x, y ) =
0, elsewhere
1
2
Z
Z Z
f (x, y )dxdy =
1
4
A
=
1
4
y
2
1
2
Z
10x y dxdy =
0
1
2
Z
Z
h
ix=y
5 x2 y2
dy
1
4
x=0
1
31
5 y 4 dy = y 5 21 =
1024
4
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Some News Continuous Joint PD Continuous Marginal Distributions
Conditional Probabilities Conditional Probability D
Example 2(b): Calculate P( 14 ≤ Y ≤ 12 )
10 x y 2 , 0 < x < y < 1
f (x, y ) =
0, elsewhere
1
4
Area A: 0 ≤ x ≤ y ≤ 1 and
1
2
Z
Z Z
f (x, y )dxdy =
1
4
A
=
1
4
y
2
1
2
1
2
Z
10x y dxdy =
0
1
2
Z
Z
≤y ≤
h
ix=y
5 x2 y2
dy
1
4
x=0
1
31
5 y 4 dy = y 5 21 =
1024
4
beamer-tu-log
Some News Continuous Joint PD Continuous Marginal Distributions
Conditional Probabilities Conditional Probability D
Example 2(b): Calculate P( 14 ≤ Y ≤ 12 )
10 x y 2 , 0 < x < y < 1
f (x, y ) =
0, elsewhere
1
4
Area A: 0 ≤ x ≤ y ≤ 1 and
1
2
Z
Z Z
f (x, y )dxdy =
1
4
A
=
1
4
y
2
1
2
1
2
Z
10x y dxdy =
0
1
2
Z
Z
≤y ≤
h
ix=y
5 x2 y2
dy
1
4
x=0
1
31
5 y 4 dy = y 5 21 =
1024
4
beamer-tu-log
Some News Continuous Joint PD Continuous Marginal Distributions
Conditional Probabilities Conditional Probability D
Example 2(c): Calculate P( 18 ≤ X ≤ 38, 14 ≤ Y ≤ 21 )
10 x y 2 , 0 < x < y < 1
f (x, y ) =
0, elsewhere
Area A
0 ≤ x ≤ y,
If 81 ≤ x ≤
x ≤ y ≤ 12
1
8
1
4,
≤ x ≤ 83 ,
1
4
then
1
4
Z
Z Z
f (x, y )dxdy =
A
=
1
8
1
2
1
2
1
2
and if
1
4
10x y 2 dy dx +
≤ x ≤ 83 , then
10
x y3
3
3
8
Z
1
4
1
4
Z
≤y ≤
≤y ≤
Z
1
8
1
4
Z
1
4
1
3
8
dx +
y = 14
10x y 2 dy dx
x
Z
2
1
2
1
4
10
x y3
3
1
2
dx
y =x
beamer-tu-log
Some News Continuous Joint PD Continuous Marginal Distributions
Conditional Probabilities Conditional Probability D
Example 2(c): Calculate P( 18 ≤ X ≤ 38, 14 ≤ Y ≤ 21 )
10 x y 2 , 0 < x < y < 1
f (x, y ) =
0, elsewhere
Area A
0 ≤ x ≤ y,
If 81 ≤ x ≤
x ≤ y ≤ 12
1
8
1
4,
≤ x ≤ 83 ,
1
4
then
1
4
Z
Z Z
f (x, y )dxdy =
A
=
1
8
1
2
1
2
1
2
and if
1
4
10x y 2 dy dx +
≤ x ≤ 83 , then
10
x y3
3
3
8
Z
1
4
1
4
Z
≤y ≤
≤y ≤
Z
1
8
1
4
Z
1
4
1
3
8
dx +
y = 14
10x y 2 dy dx
x
Z
2
1
2
1
4
10
x y3
3
1
2
dx
y =x
beamer-tu-log
Some News Continuous Joint PD Continuous Marginal Distributions
Conditional Probabilities Conditional Probability D
Example 2(c): Calculate P( 18 ≤ X ≤ 38, 14 ≤ Y ≤ 21 )
10 x y 2 , 0 < x < y < 1
f (x, y ) =
0, elsewhere
Area A
0 ≤ x ≤ y,
If 81 ≤ x ≤
x ≤ y ≤ 12
1
8
1
4,
≤ x ≤ 83 ,
1
4
then
1
4
Z
Z Z
f (x, y )dxdy =
A
=
1
8
1
2
1
2
1
2
and if
1
4
10x y 2 dy dx +
≤ x ≤ 83 , then
10
x y3
3
3
8
Z
1
4
1
4
Z
≤y ≤
≤y ≤
Z
1
8
1
4
Z
1
4
1
3
8
dx +
y = 14
10x y 2 dy dx
x
Z
2
1
2
1
4
10
x y3
3
1
2
dx
y =x
beamer-tu-log
Some News Continuous Joint PD Continuous Marginal Distributions
Conditional Probabilities Conditional Probability D
Example 2(c): Calculate P( 18 ≤ X ≤ 38, 14 ≤ Y ≤ 21 )
10 x y 2 , 0 < x < y < 1
f (x, y ) =
0, elsewhere
Area A
0 ≤ x ≤ y,
If 81 ≤ x ≤
x ≤ y ≤ 12
1
8
1
4,
≤ x ≤ 83 ,
1
4
then
1
4
Z
Z Z
f (x, y )dxdy =
A
=
1
8
1
2
1
2
1
2
and if
1
4
10x y 2 dy dx +
≤ x ≤ 83 , then
10
x y3
3
3
8
Z
1
4
1
4
Z
≤y ≤
≤y ≤
Z
1
8
1
4
Z
1
4
1
3
8
dx +
y = 14
10x y 2 dy dx
x
Z
2
1
2
1
4
10
x y3
3
1
2
dx
y =x
beamer-tu-log
Some News Continuous Joint PD Continuous Marginal Distributions
Conditional Probabilities Conditional Probability D
Example 2(c): Calculate P( 18 ≤ X ≤ 38, 14 ≤ Y ≤ 21 )
10 x y 2 , 0 < x < y < 1
f (x, y ) =
0, elsewhere
Area A
0 ≤ x ≤ y,
If 81 ≤ x ≤
x ≤ y ≤ 12
1
8
1
4,
≤ x ≤ 83 ,
1
4
then
1
4
Z
Z Z
f (x, y )dxdy =
A
=
1
8
1
2
1
2
1
2
and if
1
4
10x y 2 dy dx +
≤ x ≤ 83 , then
10
x y3
3
3
8
Z
1
4
1
4
Z
≤y ≤
≤y ≤
Z
1
8
1
4
Z
1
4
1
3
8
dx +
y = 14
10x y 2 dy dx
x
Z
2
1
2
1
4
10
x y3
3
1
2
dx
y =x
beamer-tu-log
Some News Continuous Joint PD Continuous Marginal Distributions
Conditional Probabilities Conditional Probability D
Example 2(c): Calculate P( 18 ≤ X ≤ 38, 14 ≤ Y ≤ 21 )
10 x y 2 , 0 < x < y < 1
f (x, y ) =
0, elsewhere
1
4
Z
Z Z
f (x, y )dxdy =
1
8
A
1
2
Z
3
8
Z
2
10x y dy dx +
1
4
1
4
1
2
10x y 2 dy dx
x
1
2
10
3
xy
=
dx +
dx
1
1
3
y = 41
y =x
8
4
Z 1
Z 3
4 35
8
5
10 4
x dx +
x−
x dx
=
1
1
64
12
3
8
4
3
35 h 2 i 41
5 2 2 5 8
1219
x
+
x − x
=
=
1
128
24
3
49152beamer-tu-log
x= 8
x= 1
Z
1
4
10
x y3
3
1
Z
2
Z
3
8
4
Some News Continuous Joint PD Continuous Marginal Distributions
Conditional Probabilities Conditional Probability D
And now . . .
1
Some News
2
Continuous Joint PD
3
Continuous Marginal Distributions
4
Conditional Probabilities
5
Conditional Probability Distributions
Discrete
Continuous = Density
beamer-tu-log
Some News Continuous Joint PD Continuous Marginal Distributions
Conditional Probabilities Conditional Probability D
Probability density g(x)
Let X and Y be RV’s with joint PDF f (x, y ). Then
g(x̃) = G0 (x̃) = lim
δ↓0
P(x̃ ≤ X ≤ x̃ + δ)
δ
P(x̃ ≤ X ≤ x̃ + δ)
P(x̃ ≤ X ≤ x̃ + δ) = P(x̃ ≤ X ≤ x̃ + δ, Y ∈ R)
Z ∞ Z x̃+δ
Z ∞
Z ∞
=
δ f (x̃, y ) dy = δ
f (x̃, y ) dy
f (x, y )dx dy ≈
−∞ x=x̃
−∞
−∞
{z
}
|
≈δf (x̃,y ) if δ↓0
beamer-tu-log
Some News Continuous Joint PD Continuous Marginal Distributions
Conditional Probabilities Conditional Probability D
Probability density g(x)
Let X and Y be RV’s with joint PDF f (x, y ). Then
g(x̃) = G0 (x̃) = lim
δ↓0
P(x̃ ≤ X ≤ x̃ + δ)
δ
P(x̃ ≤ X ≤ x̃ + δ)
P(x̃ ≤ X ≤ x̃ + δ) = P(x̃ ≤ X ≤ x̃ + δ, Y ∈ R)
Z ∞ Z x̃+δ
Z ∞
Z ∞
=
δ f (x̃, y ) dy = δ
f (x̃, y ) dy
f (x, y )dx dy ≈
−∞ x=x̃
−∞
−∞
{z
}
|
≈δf (x̃,y ) if δ↓0
beamer-tu-log
Some News Continuous Joint PD Continuous Marginal Distributions
Conditional Probabilities Conditional Probability D
Marginal probability density function of X - g(x)
Z ∞
P(x ≤ X ≤ x + δ)
g(x) = lim
=
f (x, y ) dy
δ
δ↓0
−∞
We integrate over the other variable.
Marginal probability density function of Y - h(x)
Z ∞
P(y ≤ Y ≤ y + )
h(y ) = lim
=
f (x, y ) dx
↓0
−∞
We integrate over the other variable.
beamer-tu-log
Some News Continuous Joint PD Continuous Marginal Distributions
Conditional Probabilities Conditional Probability D
Marginal probability density function of X - g(x)
Z ∞
P(x ≤ X ≤ x + δ)
g(x) = lim
=
f (x, y ) dy
δ
δ↓0
−∞
We integrate over the other variable.
Marginal probability density function of Y - h(x)
Z ∞
P(y ≤ Y ≤ y + )
h(y ) = lim
=
f (x, y ) dx
↓0
−∞
We integrate over the other variable.
beamer-tu-log
Some News Continuous Joint PD Continuous Marginal Distributions
Conditional Probabilities Conditional Probability D
Marginal probability density function of X - g(x)
Z ∞
P(x ≤ X ≤ x + δ)
g(x) = lim
=
f (x, y ) dy
δ
δ↓0
−∞
We integrate over the other variable.
Marginal probability density function of Y - h(x)
Z ∞
P(y ≤ Y ≤ y + )
h(y ) = lim
=
f (x, y ) dx
↓0
−∞
We integrate over the other variable.
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Some News Continuous Joint PD Continuous Marginal Distributions
Conditional Probabilities Conditional Probability D
Marginal probability density function of X - g(x)
Z ∞
P(x ≤ X ≤ x + δ)
g(x) = lim
=
f (x, y ) dy
δ
δ↓0
−∞
We integrate over the other variable.
Marginal probability density function of Y - h(x)
Z ∞
P(y ≤ Y ≤ y + )
h(y ) = lim
=
f (x, y ) dx
↓0
−∞
We integrate over the other variable.
beamer-tu-log
Some News Continuous Joint PD Continuous Marginal Distributions
Conditional Probabilities Conditional Probability D
Marginal probability density function of X - g(x)
Z ∞
P(x ≤ X ≤ x + δ)
g(x) = lim
=
f (x, y ) dy
δ
δ↓0
−∞
We integrate over the other variable.
Example 1
f (x, y ) =
y
,
2 x2
0,
1
2
< x < 1, 0 < y < 2
otherwise
Find marginal density functions g(x), h(y )
beamer-tu-log
Some News Continuous Joint PD Continuous Marginal Distributions
Conditional Probabilities Conditional Probability D
Marginal probability density function of X - g(x)
Z ∞
P(x ≤ X ≤ x + δ)
g(x) = lim
=
f (x, y ) dy
δ
δ↓0
−∞
We integrate over the other variable.
Example 1
f (x, y ) =
y
,
2 x2
0,
1
2
< x < 1, 0 < y < 2
otherwise
Find marginal density functions g(x), h(y )
beamer-tu-log
Some News Continuous Joint PD Continuous Marginal Distributions
Conditional Probabilities Conditional Probability D
Marginal probability density function of X - g(x)
Z ∞
P(x ≤ X ≤ x + δ)
=
g(x) = lim
f (x, y ) dy
δ
δ↓0
−∞
We integrate over the other variable.
Example 1
f (x, y ) =
y
,
2 x2
0,
1
2
< x < 1, 0 < y < 2
otherwise
Find marginal
density functions g(x), h(y )

h 2 i2
 R2 y
y
= x12 , 12 < x < 1
dy
=
0 2 x2
4 x 2 y =0
g(x) =
 0,
elsewhere
beamer-tu-log
Some News Continuous Joint PD Continuous Marginal Distributions
Conditional Probabilities Conditional Probability D
Marginal probability density function of X - g(x)
Z ∞
P(x ≤ X ≤ x + δ)
=
g(x) = lim
f (x, y ) dy
δ
δ↓0
−∞
We integrate over the other variable.
Example 1
f (x, y ) =
y
,
2 x2
0,
1
2
< x < 1, 0 < y < 2
otherwise
Find marginal
density functions g(x), h(y )

h 2 i2
R
 2 y
y
dy
=
= x12 , 12 < x < 1
0 2 x2
4 x 2 y =0
g(x) =
 0,
elsewhere
( R
y 1
1 y
1
0<y <2
1
2 dx = − 2 x x= 1 = 2 y ,
2
x
2
h(y ) =
2
0,
elsewhere
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Some News Continuous Joint PD Continuous Marginal Distributions
Conditional Probabilities Conditional Probability D
Marginal probability density function of X - g(x)
Z ∞
P(x ≤ X ≤ x + δ)
g(x) = lim
=
f (x, y ) dy
δ
δ↓0
−∞
Hence: Integrate over the other variable.
Example 2
f (x, y ) =
10 xy 2 , 0 < x < y < 1
0, otherwise
Find marginal density functions g(x), h(y )
beamer-tu-log
Some News Continuous Joint PD Continuous Marginal Distributions
Conditional Probabilities Conditional Probability D
Marginal probability density function of X - g(x)
Z ∞
P(x ≤ X ≤ x + δ)
g(x) = lim
=
f (x, y ) dy
δ
δ↓0
−∞
Hence: Integrate over the other variable.
Example 2
f (x, y ) =
10 xy 2 , 0 < x < y < 1
0, otherwise
Find marginal density functions g(x), h(y )
beamer-tu-log
Some News Continuous Joint PD Continuous Marginal Distributions
Conditional Probabilities Conditional Probability D
Marginal probability density function of X - g(x)
Z ∞
P(x ≤ X ≤ x + δ)
f (x, y ) dy
g(x) = lim
=
δ
δ↓0
−∞
Hence: Integrate over the other variable.
Example 2
f (x, y ) =
10 xy 2 , 0 < x < y < 1
0, otherwise
Find marginal
g(x), h(y )
 R density functions
10 3 1
1
2

 x 10 xy dy = 3 x y y =x = 10
3 x −
g(x) =
0<x <1


0,
elsewhere
10 4
3 x ,
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Some News Continuous Joint PD Continuous Marginal Distributions
Conditional Probabilities Conditional Probability D
Marginal probability density function of X - g(x)
Z ∞
P(x ≤ X ≤ x + δ)
g(x) = lim
=
f (x, y ) dy
δ
δ↓0
−∞
Hence: Integrate over the other variable.
Example 2
f (x, y ) =
10 xy 2 , 0 < x < y < 1
0, otherwise
Find marginal
g(x), h(y )
 R density functions
10 3 1
1
10 4
2

 x 10 xy dy = 3 x y y =x = 10
3 x − 3 x ,
g(x) =
0<x <1


0,
elsewhere
Ry
y
2
10 x y dx = 5 x 2 y 2 x=0 = 5 y 4 , 0 < y < 1
0
h(y ) =
0,
elsewhere
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Some News Continuous Joint PD Continuous Marginal Distributions
Conditional Probabilities Conditional Probability D
And now . . .
1
Some News
2
Continuous Joint PD
3
Continuous Marginal Distributions
4
Conditional Probabilities
5
Conditional Probability Distributions
Discrete
Continuous = Density
beamer-tu-log
Some News Continuous Joint PD Continuous Marginal Distributions
Conditional Probabilities Conditional Probability D
book: Section 2.6
Main Question
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Some News Continuous Joint PD Continuous Marginal Distributions
Conditional Probabilities Conditional Probability D
book: Section 2.6
Main Question
If we already know that an event A occurs, what is then the
probability that an event B occurs?
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Some News Continuous Joint PD Continuous Marginal Distributions
Conditional Probabilities Conditional Probability D
book: Section 2.6
Main Question
If we already know that an event A occurs, what is then the
probability that an event B occurs?
Example
S = {1, 2, 3, 4}, P({i}) =
i
10 ,
i = 1, 2, 3, 4
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Some News Continuous Joint PD Continuous Marginal Distributions
Conditional Probabilities Conditional Probability D
book: Section 2.6
Main Question
If we already know that an event A occurs, what is then the
probability that an event B occurs?
Example
S = {1, 2, 3, 4}, P({i}) =
A = {1, 2}, B = {2, 3}
i
10 ,
i = 1, 2, 3, 4
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Some News Continuous Joint PD Continuous Marginal Distributions
Conditional Probabilities Conditional Probability D
book: Section 2.6
Main Question
If we already know that an event A occurs, what is then the
probability that an event B occurs?
Example
i
, i = 1, 2, 3, 4
S = {1, 2, 3, 4}, P({i}) = 10
A = {1, 2}, B = {2, 3}
B|A : The event B given A
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Some News Continuous Joint PD Continuous Marginal Distributions
Conditional Probabilities Conditional Probability D
book: Section 2.6
Main Question
If we already know that an event A occurs, what is then the
probability that an event B occurs?
Example
i
, i = 1, 2, 3, 4
S = {1, 2, 3, 4}, P({i}) = 10
A = {1, 2}, B = {2, 3}
B|A : The event B given A = {2} = A ∩ B
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Some News Continuous Joint PD Continuous Marginal Distributions
Conditional Probabilities Conditional Probability D
book: Section 2.6
Main Question
If we already know that an event A occurs, what is then the
probability that an event B occurs?
Example
i
, i = 1, 2, 3, 4
S = {1, 2, 3, 4}, P({i}) = 10
A = {1, 2}, B = {2, 3}
B|A : The event B given A = {2} = A ∩ B
But given that A occurs the sample space is actually reduced to
A = {1, 2}
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Some News Continuous Joint PD Continuous Marginal Distributions
Conditional Probabilities Conditional Probability D
book: Section 2.6
Main Question
If we already know that an event A occurs, what is then the
probability that an event B occurs?
Example
i
S = {1, 2, 3, 4}, P({i}) = 10
, i = 1, 2, 3, 4
A = {1, 2}, B = {2, 3}
B|A : The event B given A = {2} = A ∩ B
But given that A occurs the sample space is actually reduced to
A = {1, 2}
P(B|A) =
P({2})
P({1,2})
=
2
10
3
10
=
2
3
(conditional probability)
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Some News Continuous Joint PD Continuous Marginal Distributions
Conditional Probabilities Conditional Probability D
Conditional probability
Dependence of events
In the example we have P(B) = 12 , P(B|A) =
2
3
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Some News Continuous Joint PD Continuous Marginal Distributions
Conditional Probabilities Conditional Probability D
Conditional probability
Probability of event B given A is
P(B|A) =
P(A ∩ B)
P(A)
for any two events A and B, as long as P(A) > 0.
Dependence of events
In the example we have P(B) = 12 , P(B|A) =
2
3
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Some News Continuous Joint PD Continuous Marginal Distributions
Conditional Probabilities Conditional Probability D
Conditional probability
Probability of event B given A is
P(B|A) =
P(A ∩ B)
P(A)
for any two events A and B, as long as P(A) > 0.
Dependence of events
In the example we have P(B) = 12 , P(B|A) =
2
3
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Some News Continuous Joint PD Continuous Marginal Distributions
Conditional Probabilities Conditional Probability D
Conditional probability
Probability of event B given A is
P(B|A) =
P(A ∩ B)
P(A)
for any two events A and B, as long as P(A) > 0.
Dependence of events
In the example we have P(B) = 12 , P(B|A) =
2
3
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Some News Continuous Joint PD Continuous Marginal Distributions
Conditional Probabilities Conditional Probability D
Conditional probability
Probability of event B given A is
P(B|A) =
P(A ∩ B)
P(A)
for any two events A and B, as long as P(A) > 0.
Dependence of events
In the example we have P(B) = 12 , P(B|A) = 23
So apparently the probability that B occurs changes if we have
the additional info that A occurs: P(B) 6= P(B|A)
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Some News Continuous Joint PD Continuous Marginal Distributions
Conditional Probabilities Conditional Probability D
Conditional probability
Probability of event B given A is
P(B|A) =
P(A ∩ B)
P(A)
for any two events A and B, as long as P(A) > 0.
Dependence of events
In the example we have P(B) = 12 , P(B|A) = 23
So apparently the probability that B occurs changes if we have
the additional info that A occurs: P(B) 6= P(B|A)
A and B are dependent
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Some News Continuous Joint PD Continuous Marginal Distributions
Conditional Probabilities Conditional Probability D
Independence of RV
Since P(A ∩ B) = P(A) · P(B|A) and P(A ∩ B) = P(B) · P(A|B)),
A and B are independent ⇔ P(A ∩ B) = P(A) · P(B)
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Some News Continuous Joint PD Continuous Marginal Distributions
Conditional Probabilities Conditional Probability D
Independence of RV
A and B are called independent if (a) P(A|B) = P(A) or (b)
P(B|A) = P(B)
Since P(A ∩ B) = P(A) · P(B|A) and P(A ∩ B) = P(B) · P(A|B)),
A and B are independent ⇔ P(A ∩ B) = P(A) · P(B)
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Some News Continuous Joint PD Continuous Marginal Distributions
Conditional Probabilities Conditional Probability D
Independence of RV
A and B are called independent if (a) P(A|B) = P(A) or (b)
P(B|A) = P(B)
(a) and (b) are equivalent if P(A) > 0, P(B) > 0
Since P(A ∩ B) = P(A) · P(B|A) and P(A ∩ B) = P(B) · P(A|B)),
A and B are independent ⇔ P(A ∩ B) = P(A) · P(B)
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Some News Continuous Joint PD Continuous Marginal Distributions
Conditional Probabilities Conditional Probability D
Independence of RV
A and B are called independent if (a) P(A|B) = P(A) or (b)
P(B|A) = P(B)
(a) and (b) are equivalent if P(A) > 0, P(B) > 0
Since P(A ∩ B) = P(A) · P(B|A) and P(A ∩ B) = P(B) · P(A|B)),
A and B are independent ⇔ P(A ∩ B) = P(A) · P(B)
beamer-tu-log
Some News Continuous Joint PD Continuous Marginal Distributions
Conditional Probabilities Conditional Probability D
And now . . .
1
Some News
2
Continuous Joint PD
3
Continuous Marginal Distributions
4
Conditional Probabilities
5
Conditional Probability Distributions
Discrete
Continuous = Density
beamer-tu-log
Some News Continuous Joint PD Continuous Marginal Distributions
Conditional Probabilities Conditional Probability D
Discrete
book: Section 3.4
Conditional Probability Distribution
Let X and Y be discrete RV’s with joint probability distribution f .
The conditional probability of X given that Y has the value y is
(x,y )
=x,Y =y )
= f h(y
f (x|y ) = P(X = x|Y = y ) = P(XP(Y
=y )
)
Example
Throwing 2 dice, X : maximum, Y : minimum. What is
conditional probability distribution of X given Y = 2?
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Some News Continuous Joint PD Continuous Marginal Distributions
Conditional Probabilities Conditional Probability D
Discrete
book: Section 3.4
Conditional Probability Distribution
Let X and Y be discrete RV’s with joint probability distribution f .
The conditional probability of X given that Y has the value y is
(x,y )
=x,Y =y )
= f h(y
f (x|y ) = P(X = x|Y = y ) = P(XP(Y
=y )
)
Example
Throwing 2 dice, X : maximum, Y : minimum. What is
conditional probability distribution of X given Y = 2?
beamer-tu-log
Some News Continuous Joint PD Continuous Marginal Distributions
Conditional Probabilities Conditional Probability D
Discrete
book: Section 3.4
Conditional Probability Distribution
Let X and Y be discrete RV’s with joint probability distribution f .
The conditional probability of X given that Y has the value y is
(x,y )
=x,Y =y )
= f h(y
f (x|y ) = P(X = x|Y = y ) = P(XP(Y
=y )
)
Example
Throwing 2 dice, X : maximum, Y : minimum. What is
conditional probability distribution of X given Y = 2?
1
(CAH) P(Y = 2) = 14 , x = 3, 4, 5, 6 : f (x, y ) = 18
,
1
x = 2 : f (x, y ) = 36
beamer-tu-log
Some News Continuous Joint PD Continuous Marginal Distributions
Conditional Probabilities Conditional Probability D
Discrete
book: Section 3.4
Conditional Probability Distribution
Let X and Y be discrete RV’s with joint probability distribution f .
The conditional probability of X given that Y has the value y is
(x,y )
=x,Y =y )
= f h(y
f (x|y ) = P(X = x|Y = y ) = P(XP(Y
=y )
)
Example
Throwing 2 dice, X : maximum, Y : minimum. What is
conditional probability distribution of X given Y = 2?
1
(CAH) P(Y = 2) = 14 , x = 3, 4, 5, 6 : f (x, y ) = 18
,
x
2
3, 4, 5
1
1
1
x = 2 : f (x, y ) = 36
P(X = x|Y = 2) 361 = 91 181 = 92
4
4
beamer-tu-log
Some News Continuous Joint PD Continuous Marginal Distributions
Conditional Probabilities Conditional Probability D
Continuous = Density
book: Section 3.4
Conditional Probability Distribution
Let X and Y be continuous RV’s with joint probability density f .
The conditional probability of X given that Y has the value y is
P(x ≤ X ≤ x + δ|y ≤ Y ≤ y + )
δ
(δ,)↓0
P(x ≤ X ≤ x + δ, y ≤ Y ≤ y + )
= lim
δ · P(y ≤ Y ≤ y + )
(δ,)↓0
P(x ≤ X ≤ x + δ, y ≤ Y ≤ y + )
= lim
·
δ
P(y ≤ Y ≤ y + )
(δ,)↓0
1
f (x, y )
= f (x, y ) ·
=
, provided h(y ) > 0
h(y )
h(y )
f (x|y ) = lim
Same formula as for discrete case!
beamer-tu-log
Some News Continuous Joint PD Continuous Marginal Distributions
Conditional Probabilities Conditional Probability D
Continuous = Density
book: Section 3.4
Conditional Probability Distribution
Let X and Y be continuous RV’s with joint probability density f .
The conditional probability of X given that Y has the value y is
P(x ≤ X ≤ x + δ|y ≤ Y ≤ y + )
δ
(δ,)↓0
P(x ≤ X ≤ x + δ, y ≤ Y ≤ y + )
= lim
δ · P(y ≤ Y ≤ y + )
(δ,)↓0
P(x ≤ X ≤ x + δ, y ≤ Y ≤ y + )
= lim
·
δ
P(y ≤ Y ≤ y + )
(δ,)↓0
1
f (x, y )
= f (x, y ) ·
=
, provided h(y ) > 0
h(y )
h(y )
f (x|y ) = lim
Same formula as for discrete case!
beamer-tu-log
Some News Continuous Joint PD Continuous Marginal Distributions
Conditional Probabilities Conditional Probability D
Continuous = Density
book: Section 3.4
Conditional Probability Distribution
Let X and Y be continuous RV’s with joint probability density f .
The conditional probability of X given that Y has the value y is
P(x ≤ X ≤ x + δ|y ≤ Y ≤ y + )
δ
(δ,)↓0
P(x ≤ X ≤ x + δ, y ≤ Y ≤ y + )
= lim
δ · P(y ≤ Y ≤ y + )
(δ,)↓0
P(x ≤ X ≤ x + δ, y ≤ Y ≤ y + )
= lim
·
δ
P(y ≤ Y ≤ y + )
(δ,)↓0
1
f (x, y )
= f (x, y ) ·
=
, provided h(y ) > 0
h(y )
h(y )
f (x|y ) = lim
Same formula as for discrete case!
beamer-tu-log
Some News Continuous Joint PD Continuous Marginal Distributions
Conditional Probabilities Conditional Probability D
Continuous = Density
Example - from before
Let X and Y be continuous RV’s with joint probability density f
10 x y 2 , , 0 < x < y ≤ 1
f (x, y ) =
0,
elsewhere
Calculate conditional density of X , given Y =
P(X > 13 |Y = 12 )
1
2
and calculate
beamer-tu-log
Some News Continuous Joint PD Continuous Marginal Distributions
Conditional Probabilities Conditional Probability D
Continuous = Density
Example - from before
Let X and Y be continuous RV’s with joint probability density f
10 x y 2 , , 0 < x < y ≤ 1
f (x, y ) =
0,
elsewhere
Calculate conditional density of X , given Y = 21 and calculate
P(X > 13 |Y = 12 )
10 4
4
y (x) = 10
3 x − 3 x , 0 < x < 1, h(y ) = 5 y , 0 < y < 1
beamer-tu-log
Some News Continuous Joint PD Continuous Marginal Distributions
Conditional Probabilities Conditional Probability D
Continuous = Density
Example - from before
Let X and Y be continuous RV’s with joint probability density f
10 x y 2 , , 0 < x < y ≤ 1
f (x, y ) =
0,
elsewhere
Calculate conditional density of X , given Y = 21 and calculate
P(X > 13 |Y = 12 )
10 4
4
y (x) = 10
3 x − 3 x , 0 < x < 1, h(y ) = 5 y , 0 < y < 1
f (x|y ) =
10 x y 2
5 y4
=
2x
,
y2
0<x <y <1
beamer-tu-log
Some News Continuous Joint PD Continuous Marginal Distributions
Conditional Probabilities Conditional Probability D
Continuous = Density
Example - from before
Let X and Y be continuous RV’s with joint probability density f
10 x y 2 , , 0 < x < y ≤ 1
f (x, y ) =
0,
elsewhere
Calculate conditional density of X , given Y = 21 and calculate
P(X > 13 |Y = 12 )
10 4
4
y (x) = 10
3 x − 3 x , 0 < x < 1, h(y ) = 5 y , 0 < y < 1
f (x|y ) =
f (x| 12 ) =
10 x y 2
= 2x
,0<x <y <1
y2
h 5 y i4
2x
= 8 x, 0 < x < 12
y2 y= 1
2
beamer-tu-log
Some News Continuous Joint PD Continuous Marginal Distributions
Conditional Probabilities Conditional Probability D
Continuous = Density
Example - from before
Let X and Y be continuous RV’s with joint probability density f
10 x y 2 , , 0 < x < y ≤ 1
f (x, y ) =
0,
elsewhere
Calculate conditional density of X , given Y = 21 and calculate
P(X > 13 |Y = 12 )
10 4
4
y (x) = 10
3 x − 3 x , 0 < x < 1, h(y ) = 5 y , 0 < y < 1
f (x|y ) =
f (x| 12 ) =
10 x y 2
= 2x
,0<x <y <1
y2
h 5 y i4
2x
= 8 x, 0 < x < 12
y2 y= 1
2
P(X > 13 |Y = 12 ) =
R
1
2
1
3
1
8 xdx = 4 x 2 2
x= 31
=
5
9
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