Download Jointly distributed Random variables

Jointly distributed Random
variables
Multivariate distributions
Quite often there will be 2 or more random
variables (X, Y, etc) defined for the same
random experiment.
Example:
A bridge hand (13 cards) is selected from a deck
of 52 cards.
X = the number of spades in the hand.
Y = the number of hearts in the hand.
In this example we will define:
p(x,y) = P[X = x, Y = y]
The function
p(x,y) = P[X = x, Y = y]
is called the joint probability function of
X and Y.
Note:
The possible values of X are 0, 1, 2, …, 13
The possible values of Y are also 0, 1, 2, …, 13
and X + Y ≤ 13.
p  x, y   P  X  x, Y  y 
The number of
ways of choosing
the x spades for the
hand
The number of
ways of choosing
the y hearts for the
hand
The number of ways
of completing the hand
with diamonds and
clubs.
13 13  26 
  

x  y 13  x  y 


 52 
The total number of
 
ways of choosing the
13
 
13 cards for the hand
Table: p(x,y)
0
13 13  26 
  

x  y 13  x  y 


 52 
 
 13 
1
2
3
4
5
6
7
8
9
10
11
12
13
0 0.0000
0.0002
0.0009
0.0024
0.0035
0.0032
0.0018
0.0006
0.0001
0.0000
0.0000
0.0000
0.0000
0.0000
1 0.0002
0.0021
0.0085
0.0183
0.0229
0.0173
0.0081
0.0023
0.0004
0.0000
0.0000
0.0000
0.0000
-
2 0.0009
0.0085
0.0299
0.0549
0.0578
0.0364
0.0139
0.0032
0.0004
0.0000
0.0000
0.0000
-
-
3 0.0024
0.0183
0.0549
0.0847
0.0741
0.0381
0.0116
0.0020
0.0002
0.0000
0.0000
-
-
-
4 0.0035
0.0229
0.0578
0.0741
0.0530
0.0217
0.0050
0.0006
0.0000
0.0000
-
-
-
-
5 0.0032
0.0173
0.0364
0.0381
0.0217
0.0068
0.0011
0.0001
0.0000
-
-
-
-
-
6 0.0018
0.0081
0.0139
0.0116
0.0050
0.0011
0.0001
0.0000
-
-
-
-
-
-
7 0.0006
0.0023
0.0032
0.0020
0.0006
0.0001
0.0000
-
-
-
-
-
-
-
8 0.0001
0.0004
0.0004
0.0002
0.0000
0.0000
-
-
-
-
-
-
-
-
9 0.0000
0.0000
0.0000
0.0000
0.0000
-
-
-
-
-
-
-
-
-
10 0.0000
0.0000
0.0000
0.0000
-
-
-
-
-
-
-
-
-
-
11 0.0000
0.0000
0.0000
-
-
-
-
-
-
-
-
-
-
-
12 0.0000
0.0000
-
-
-
-
-
-
-
-
-
-
-
-
13 0.0000
-
-
-
-
-
-
-
-
-
-
-
-
-
Bar graph: p(x,y)
0.0900
p(x,y)
0.0800
0.0700
0.0600
0.0500
0.0400
0.0300
0.0200
12
0.0100
9
6
0
1
3
2
3
4
5
x
6
7
8
9
0
10 11
12 13
y
Example:
A die is rolled n = 5 times
X = the number of times a “six” appears.
Y = the number of times a “five” appears.
Now
p(x,y) = P[X = x, Y = y]
The possible values of X are 0, 1, 2, 3, 4, 5.
The possible values of Y are 0, 1, 2, 3, 4, 5.
and X + Y ≤ 5
A typical outcome of rolling a die n = 5 times
will be a sequence F5FF6 where F denotes the
outcome {1,2,3,4}. The probability of any such
sequence will be:
x
y
1 1 4
     
6 6 6
5 x  y
where
x = the number of sixes in the sequence and
y = the number of fives in the sequence
Now
p(x,y) = P[X = x, Y = y]
x
y
1 1 4
K     
6 6 6
5 x  y
Where K = the number of sequences of length 5
containing x sixes and y fives.
 5  5  x  5  x  y 
  


x
y
5

x

y
 



  5  x ! 
5!
5!




 x ! 5  x !  y ! 5  x  y !  x ! y ! 5  x  y !



Thus
p(x,y) = P[X = x, Y = y]
x
y
5!
1 1 4

     
x ! y ! 5  x  y !  6   6   6 
if x + y ≤ 5 .
5 x  y
Table: p(x,y)
x
y
5!
1 1 4

     
x ! y ! 5  x  y !  6   6   6 
0
1
2
3
4
5
0
0.1317
0.1646
0.0823
0.0206
0.0026
0.0001
1
0.1646
0.1646
0.0617
0.0103
0.0006
0
2
0.0823
0.0617
0.0154
0.0013
0
0
3
0.0206
0.0103
0.0013
0
0
0
5 x  y
4
0.0026
0.0006
0
0
0
0
5
0.0001
0
0
0
0
0
Bar graph: p(x,y)
p(x,y)
0.1800
0.1600
0.1400
0.1200
0.1000
0.0800
0.0600
0.0400
5
4
0.0200
3
2
0.0000
0
1
1
2
x
3
0
4
5
y
General properties of the joint probability
function;
p(x,y) = P[X = x, Y = y]
1.
0  p  x, y   1
2.
 p  x, y   1
x
3.
y
P  X , Y   A   p  x, y 
 x, y   A
Example:
A die is rolled n = 5 times
X = the number of times a “six” appears.
Y = the number of times a “five” appears.
What is the probability that we roll more sixes
than fives
i.e. what is P[X > Y]?
Table: p(x,y)
x
y
5!
1 1 4

     
x ! y ! 5  x  y !  6   6   6 
0
1
2
3
4
5
0
0.1317
0.1646
0.0823
0.0206
0.0026
0.0001
1
0.1646
0.1646
0.0617
0.0103
0.0006
0
2
0.0823
0.0617
0.0154
0.0013
0
0
3
0.0206
0.0103
0.0013
0
0
0
P  X  Y    p  x, y   0.3441
x y
5 x  y
4
0.0026
0.0006
0
0
0
0
5
0.0001
0
0
0
0
0
Marginal and conditional
distributions
Definition:
Let X and Y denote two discrete random variables
with joint probability function
p(x,y) = P[X = x, Y = y]
Then
pX(x) = P[X = x] is called the marginal
probability function of X.
and
pY(y) = P[Y = y] is called the marginal
probability function of Y.
Note: Let y1, y2, y3, … denote the possible values of Y.
pX  x   P  X  x 
 P  X  x, Y  y1   X  x, Y  y2  

 P  X  x, Y  y1   P  X  x, Y  y2  
 p  x, y1   p  x, y2  
  p  x, y j    p  x, y 
j
y
Thus the marginal probability function of X, pX(x) is
obtained from the joint probability function of X and Y by
summing p(x,y) over the possible values of Y.
Also
pY  y   P Y  y 
 P  X  x1 , Y  y   X  x2 , Y  y 
 P  X  x1 , Y  y   P  X  x2 , Y  y  
 p  x1 , y   p  x2 , y  
  p  xi , y    p  x, y 
i
x

Example:
A die is rolled n = 5 times
X = the number of times a “six” appears.
Y = the number of times a “five” appears.
0
1
2
3
4
5
p Y (y )
0
0.1317
0.1646
0.0823
0.0206
0.0026
0.0001
0.4019
1
0.1646
0.1646
0.0617
0.0103
0.0006
0
0.4019
2
0.0823
0.0617
0.0154
0.0013
0
0
0.1608
3
0.0206
0.0103
0.0013
0
0
0
0.0322
4
0.0026
0.0006
0
0
0
0
0.0032
5
0.0001
0
0
0
0
0
0.0001
p X (x )
0.4019
0.4019
0.1608
0.0322
0.0032
0.0001
Conditional Distributions
Definition:
Let X and Y denote two discrete random variables
with joint probability function
p(x,y) = P[X = x, Y = y]
Then
pX |Y(x|y) = P[X = x|Y = y] is called the conditional
probability function of X given Y
=y
and
pY |X(y|x) = P[Y = y|X = x] is called the conditional
probability function of Y given
X=x
Note
pX Y  x y   P  X  x Y  y 
P  X  x, Y  y  p  x , y 


P Y  y 
pY  y 
and
pY X  y x   P Y  y X  x 

P  X  x, Y  y 
P  X  x

p  x, y 
pX  x 
• Marginal distributions describe how one
variable behaves ignoring the other variable.
• Conditional distributions describe how one
variable behaves when the other variable is
held fixed
Example:
A die is rolled n = 5 times
X = the number of times a “six” appears.
Y = the number of times a “five” appears.
y
x
0
1
2
3
4
5
p Y (y )
0
0.1317
0.1646
0.0823
0.0206
0.0026
0.0001
0.4019
1
0.1646
0.1646
0.0617
0.0103
0.0006
0
0.4019
2
0.0823
0.0617
0.0154
0.0013
0
0
0.1608
3
0.0206
0.0103
0.0013
0
0
0
0.0322
4
0.0026
0.0006
0
0
0
0
0.0032
5
0.0001
0
0
0
0
0
0.0001
p X (x )
0.4019
0.4019
0.1608
0.0322
0.0032
0.0001
The conditional distribution of X given Y = y.
pX |Y(x|y) = P[X = x|Y = y]
y
x
0
1
2
3
4
5
0
0.3277
0.4096
0.2048
0.0512
0.0064
0.0003
1
0.4096
0.4096
0.1536
0.0256
0.0016
0.0000
2
0.5120
0.3840
0.0960
0.0080
0.0000
0.0000
3
0.6400
0.3200
0.0400
0.0000
0.0000
0.0000
4
0.8000
0.2000
0.0000
0.0000
0.0000
0.0000
5
1.0000
0.0000
0.0000
0.0000
0.0000
0.0000
The conditional distribution of Y given X = x.
pY |X(y|x) = P[Y = y|X = x]
y
x
0
1
2
3
4
5
0
0.3277
0.4096
0.5120
0.6400
0.8000
1.0000
1
0.4096
0.4096
0.3840
0.3200
0.2000
0.0000
2
0.2048
0.1536
0.0960
0.0400
0.0000
0.0000
3
0.0512
0.0256
0.0080
0.0000
0.0000
0.0000
4
0.0064
0.0016
0.0000
0.0000
0.0000
0.0000
5
0.0003
0.0000
0.0000
0.0000
0.0000
0.0000
Example
A Bernoulli trial (S - p, F – q = 1 – p) is repeated until
two successes have occurred.
X = trial on which the first success occurs
and
Y = trial on which the 2nd success occurs.
Find the joint probability function of X, Y.
Find the marginal probability function of X and Y.
Find the conditional probability functions of Y given X
= x and X given Y = y,
Solution
A typical outcome would be:
x
y
FFF…FSFFF…FS
x-1
y–x-1
p  x, y   P  X  x, Y  y 
 q x1 pq y  x1 p  q y 2 p 2 if y  x
q y  2 p 2
p  x, y   
 0
if y  x
otherwise
p(x,y) - Table
y
x
1
2
3
4
5
6
7
8
1
0
0
0
0
0
0
0
0
2
p2
0
0
0
0
0
0
0
3
4
5
6
p2q p2q2 p2q3 p2q4
p2q p2q2 p2q3 p2q4
0 p2q2 p2q3 p2q4
0
0 p2q3 p2q4
0
0
0 p2q4
0
0
0
0
0
0
0
0
0
0
0
0
7
p2q5
p2q5
p2q5
p2q5
p2q5
p2q5
0
0
8
p2q6
p2q6
p2q6
p2q6
p2q6
p2q6
p2q6
0
The marginal distribution of X
p X  x   P  X  x    p  x, y 

y


2
pq
y 2
y  x 1
 p 2 q x1  p 2 q x  p 2 q x1  p 2 q x2 
pq
x 1
pq
x 1
2
2
1  q  q
2
q 
3
 1 
x 1

  pq
 1 q 
This is the geometric distribution

The marginal distribution of Y
pY  y   P Y  y    p  x, y 
x
 y  1 p 2 q y 2

0

y  2,3, 4,
otherwise
This is the negative binomial distribution with k = 2.
The conditional distribution of X given Y = y
pX Y  x y   P  X  x Y  y 

P  X  x, Y  y 
P Y  y 

p  x, y 
pY  y 
p 2 q y 2

pq x 1
 pq y  x 1 for y  x  1, x  2, x  3
This is the geometric distribution with time starting at x.
The conditional distribution of Y given X = x
pY X  y x   P Y  y X  x 

P  X  x, Y  y 
P  X  x

p  x, y 
pX  x 
p 2 q y 2
1


2 y 2
 y  1 p q
 y  1
for x  1, 2,3,
,  y 1
This is the uniform distribution on the values 1, 2, …(y – 1)
Summary
Discrete Random Variables
The joint probability function;
p(x,y) = P[X = x, Y = y]
1.
0  p  x, y   1
2.
 p  x, y   1
x
3.
y
P  X , Y   A   p  x, y 
 x, y   A
Continuous Random Variables
Definition: Two random variable are said to have
joint probability density function f(x,y) if
1.
0  f  x, y 
 
2.
  f  x, y  dxdy  1
 
3.
P  X , Y   A    f  x, y  dxdy
A
If 0  f  x, y  then z  f  x, y 
defines a surface over the x – y plane
  f  x, y  dxdy
A
Multiple Integration
  f  x, y  dxdy
A
f(x,y)
If the region A = {(x,y)| a ≤ x ≤ b, c ≤ y ≤ d} is a
rectangular region with sides parallel to the
coordinate axes:
y
d
c
a
Then
x
b
  f  x, y  dxdy
A




    f  x, y  dx  dy     f  x, y  dy  dx




d
b
b
d
c
a
a
c
To evaluate
d b
  f  x, y  dxdy    f  x, y  dxdy
c a
d b
A


    f  x, y  dx  dy
c a

First evaluate the inner integral
b
G  y    f  x, y  dx
a
Then evaluate the outer integral
d b
d
c a
c
  f  x, y  dxdy   G  y  dy
y
d
dy
y
c
b
a
b
x
G  y    f  x, y  dx = area under surface above the
line where y is constant
a
d b
d
c a
c
  f  x, y  dxdy   G  y  dy
Infinitesimal volume under
surface above the line where
y is constant
The same quantity can be calculated by integrating
first with respect to y, than x.
b d
  f  x, y  dxdy    f  x, y  dydx
a c
b d
A


    f  x, y  dy  dx
a c

First evaluate the inner integral
d
H  x    f  x, y  dy
c
Then evaluate the outer integral
b d
b
a c
a
  f  x, y  dydx   H  x  dx
y
dx
d
c
d
a
x
b
x
H  x    f  x, y  dy = area under surface above the
line where x is constant
c
b d
b
a c
a
  f  x, y  dydx   H  x  dx
Infinitesimal volume under
surface above the line where
x is constant
Example: Compute
1 1
 
1 1

0 0
Now
1 1
 
0 0


x 2 y  xy 3 dxdy    x 2 y  xy 3 dydx
0 0
1


2
3
2
3
x y  xy dxdy     x y  xy dx  dy
0 0

x

1
1 3
2
x
x 3 
 dy
  y y
3
2
0
x 0 


1
2
4 y 1
1 3
1y 1y
1
   y  y  dy 

3
2 
3 2 2 4 y 0
0
1 1 7
  
6 8 24

1


The same quantity can be computed by reversing
the order of integration
1 1
 
0 0
1


2
3
2
3
x y  xy dydx     x y  xy dy  dx
0 0


1


y 1
4
 y2

y
2
   x  x  dx
2
4 y 0 
0


1
2 x 1
1x 1x
1 2 1 
   x  x  dx 

2
4 
2 3 4 2
0
1 1 7
  
6 8 24
1
3
x 0
Integration over non rectangular
regions
Suppose the region A is defined as follows
A = {(x,y)| a(y) ≤ x ≤ b(y), c ≤ y ≤ d}
y
d
c
Then

A
b(y)
a(y)
x
 b y 

f  x, y  dxdy     f  x, y  dx  dy

c 
 a y 
d
If the region A is defined as follows
A = {(x,y)| a ≤ x ≤ b, c(x) ≤ y ≤ d(x) }
y
d(x)
c(x)
Then

A
b
a
x
d  x

f  x, y  dxdy     f  x, y  dy  dx

a
 c x 
b
In general the region A can be partitioned into
regions of either type
y
A2
A1
A3
A
A4
x
Example:
Compute the volume under f(x,y) = x2y + xy3 over the
region A = {(x,y)| x + y ≤ 1, 0 ≤ x, 0 ≤ y}
y
(0, 1)
x+y=1
(1, 0)
x
Integrating first with respect to x than y
y
(0, 1)
x+y=1
(1 - y, y)
(0, y)
(1, 0)
x
 x
A
2

1 1 y
y  xy dxdy 
3
 x
0 0
1 1 y
2

y  xy dxdy
3


2
3
    x y  xy dx  dy

0 
0


and
x 1 y
3
2




x
x
2
3
3
x
y

xy
dx
dy

    3 y  2 y  dy
0  0 


0
x 0 


3
2
1
1 y
1 y 3 


 
y
y  dy
3
2

0

1
 y  3 y 2  3 y3  y 4 y3  2 y 4  y5 
 

 dy
3
2

0
1
1 y
1
 1  43  15 14  52  16


3
2
 16  13  14  151  81  15  121
1
2

20 4030815 2410
120
3
 120

1
40
Now integrating first with respect to y than x
y
(0, 1)
x+y=1
(x, 1 – x )
(1, 0)
(x, 0)
 
A

x
1 1 x
x 2 y  xy 3 dydx 

0 0
1 1 x

x 2 y  xy 3 dydx


2
3
    x y  xy dy  dx
0  0



Hence

0  0
1
1 x
y 1 x
2
4




y
y

2
2
3


x
 x y  xy  dy  dx 0  2  x 4   dx
y 0 


1
1
x
0
2
1  x 
2
2
1 x

x
4
4
dx
x 2  2 x3  x 4 x  4 x 2  6 x3  4 x 4  x5


dx
2
4
0
1
x  2 x 2  2 x3  2 x 4  x5

dx
4
0
1
1512 6
4
 18  16  18  101  201  1520120
 120
Continuous Random Variables
Definition: Two random variable are said to have
joint probability density function f(x,y) if
1.
0  f  x, y 
 
2.
  f  x, y  dxdy  1
 
3.
P  X , Y   A    f  x, y  dxdy
A
Definition: Let X and Y denote two random
variables with joint probability density function
f(x,y) then
the marginal density of X is
fX  x 

 f  x, y  dy

the marginal density of Y is
fY  y  

 f  x, y  dx

Definition: Let X and Y denote two random
variables with joint probability density function
f(x,y) and marginal densities fX(x), fY(y) then
the conditional density of Y given X = x
fY X  y x  
f  x, y 
fX  x
conditional density of X given Y = y
fX Y  x y 
f  x, y 
fY  y 
The bivariate Normal distribution
Let
f  x1 , x2  
1
 2   1 2
1 
2
e
1
 Q  x1 , x2 
2
where
2
 x   2






x


x


x


1
1
1
1
2
2
2
2

2






 
 
  1   2    2  
  1 
Q  x1 , x2  
1  2
This distribution is called the bivariate
Normal distribution.
The parameters are 1, 2 , 1, 2 and .
Surface Plots of the bivariate
Normal distribution
Note:
f  x1 , x2  
1
 2   1 2
1 
2
e
1
 Q  x1 , x2 
2
is constant when
2
 x   2






x


x


x


1
1
1
1
2
2
2
2

2






 
 
  1   2    2  
  1 
Q  x1 , x2  
1  2
is constant.
This is true when x1, x2 lie on an ellipse
centered at 1, 2 .
Marginal and Conditional
distributions
Marginal distributions for the Bivariate Normal
distribution
Recall the definition of marginal distributions
for continuous random variables:
f1  x1  



f  x1 , x2  dx2
and
f 2  x2  

 f  x , x  dx
1
2

It can be shown that in the case of the bivariate
normal distribution the marginal distribution of xi
is Normal with mean i and standard deviation i.
1
Proof:
The marginal distributions of x2 is
f 2  x2  

 f  x , x  dx
1
2
1



1
 2 1 2
1  2
e
1
 Q x1 , x2 
2
dx1

where
2

 x1  1  x2  2
 x1  1 

  2 

 1 
  1   2


Q  x1 , x2  
1  2
  x2  2  


 
  2  

2
Now:
2
 x   2






x


x


x


1
1
1
1
2
2
2
2

2






 
 
  1   2    2  
  1 
Q  x1 , x2  
1  2
2
2
x

a
x
a
a
 1

1


c

 2 2 x1  2  c

2
b
b
b
 b 
2

x12
1
x2  2
 
2 

2
2
 1  
 2 1 1   2
   1  



2
1
 1  
2
1
2


 2


 x2  2 
 2 1 1  
2

1

x2  2 



 x1

2
 22 1   2 
Hence
Also

b2   1   2

or b  1 1   2
1
x2  2
a
 

2
2
b
 1  
 2 1 1   2






1
1
 
    x2   
2  1
2
 1    

and
1
a  1  
 x2   
2
Finally
x2  2 
x2  2 



a
c  2
 2
1  2
2
2
2
b
1 1  
 2 1 1  
 2 1  2
2

c


2
1
 1  
2
1
2

2
1
 1  
2
1
2
2
1


2

 2
 2

 x2  2 
 2 1 1  
2
 x2  2 
 2 1 1  
2





1
 1     x2    
2


  1   2 
2
 x2  2 
2
x2  2 


2
1 
1

 22 1   2 
 22 1   2 

a2
 2
b
and
 2
1
12
1
2
c 2
  2 1  x2  2   2  x2  2 
2  1
2
2
1 1    
2

 
1
  1    x2  2  
2

 
 12
1
2
2
 2
1     x2  2  
2  2 
1 1      2

 x2  2 


 2 
2
Summarizing
2
 x   2






x


x


x


1
1
1
1
2
2
2
2

  2 


 
  1   2    2  
  1 
Q  x1 , x2  
1  2
 x1  a 

 c
 b 
2
where
and
b  1 1   2
1
a  1  
 x2   
2
2
 x2   2 
c

 2 
Thus

f 2  x2  
 f  x , x  dx
1
2
1



1
 2 1 2



1 
e
2
1
2 be
 2 1 2
1
2 2
e
1  2
e
1  2
2
2

1  x1  a 
 
c
2  b 

dx1


c 2
1  x  
  2 2
2 2 
dx1


 2 1 2
1
 Q x1 , x2 
2


1
e
2 b
1  x a 
  1 
2 b 
2
dx1
Thus the marginal distribution of x2 is Normal
with mean 2 and standard deviation 2.
Similarly the marginal distribution of x1 is Normal
with mean 1 and standard deviation 1.
Conditional distributions for the Bivariate Normal
distribution
Recall the definition of conditional distributions
for continuous random variables:
f1 2  x1 x2  
f  x1 , x2 
f 2  x2 
and f 21  x2 x1  
f  x1 , x2 
f1  x1 
It can be shown that in the case of the bivariate
normal distribution the conditional distribution of
xi given xj is Normal with:
i
mean i j  i    x j   j  and
j
standard deviation
i j  i 1  2
Proof
f 21  x2 x1  
f  x1 , x2 
f1  x1 
e
1
 Q  x1 , x2 
2
2   1 2


1  2
1
2 2

e
1
1  x  
 Q  x1 , x2    2 2 
2
2 2 
2 1 1  
2
2

e
e
1  x  
  2 2
2 2 
2
2
 1  x   2
1  x1  a 
 
c   2 2 

2  b 
 2   2 
2 1 1   2
where
and
Hence
b  1 1   2
1
a  1  
 x2   
2
2
 x2   2 
c

 2 
1
f1 2  x1 x2  
e
2 b
1  x a 
  1 
2 b 
2
Thus the conditional distribution of x2 given x1 is Normal
with:
1
 x2  2  and
mean a  1 2  1  
2
standard deviation
b  1 2  1 1   2
Bivariate Normal Distribution with marginal
distributions
Bivariate Normal Distribution with
conditional distribution
x2
( 1, 2)
Major axis of
ellipses
Regression
Regression to the
mean
2
21  2    x1  2 
1
x1
Example:
Suppose that a rectangle is constructed by first choosing
its length, X and then choosing its width Y.
Its length X is selected form an exponential distribution
with mean  = 1/l = 5. Once the length has been chosen
its width, Y, is selected from a uniform distribution form
0 to half its length.
Find the probability that its area A = XY is less than 4.
Solution:
fX  x  e
1
5
fY X
 15 x
for x  0
1
 y x   x 2 if 0  y  x 2
f  x, y   f X  x  fY X  y x 
 e
1
5
 15 x
1
1
2 5 x
= 5x e
if 0  y  x 2, x  0
x2
y x 24 x
x 2  8 or x  8  2 2
yx 22 2 2 2
xy = 4
2
2, 2
y = x/2

P  XY  4 
x
2 2 2
4
x
  f  x, y  dydx    f  x, y  dydx
0
2

2, 2
0

2 2 0
P  XY  4 
x
2 2 2
0
 
0
2
5x
e
 15 x

2
5x
e

0
This part can be
evaluated
2
5x
e
 15 x
dydx
 15 x

x
2
dx 

2
5x
e
 15 x 4
x
dx
2 2
2 2

x
2 2 0
0
1
5
4
 
dydx 
0
2 2

x
2 2 0
x
2 2 2

4
  f  x, y  dydx    f  x, y  dydx
0


e
 15 x

dx 
8
5

2  15 x
x e
dx
2 2
This part may require
Numerical evaluation
multivariate distributions
k≥2
Definition
Let X1, X2, …, Xk denote k discrete random
variables, then
p(x1, x2, …, xk )
is joint probability function of X1, X2, …, Xk if
1.
0  p  x1 ,
2.
  px ,
1
x1
3.
, xn   1
, xn   1
xn
P  X 1 ,
, X n   A  
 x1,
 px ,
1
, xn   A
, xn 
Definition
Let X1, X2, …, Xk denote k continuous random
variables, then
f(x1, x2, …, xk )
is joint density function of X1, X2, …, Xk if
1.
f  x1 ,

2.

  f x ,
, xn  dx1 ,
, dxn  1
, X n   A  
 f x ,
1

3.
, xn   0

P  X 1 ,
1
A
, xn  dx1 ,
, dxn
Example: The Multinomial distribution
Suppose that we observe an experiment that has k
possible outcomes {O1, O2, …, Ok } independently n
times.
Let p1, p2, …, pk denote probabilities of O1, O2, …,
Ok respectively.
Let Xi denote the number of times that outcome Oi
occurs in the n repetitions of the experiment.
Then the joint probability function of the random
variables X1, X2, …, Xk is
n!
p  x1 , , xn  
p1x p2x
pkx
x1 ! x2 ! xk !
1
2
k
Note:
p1x1 p2x2
pkxk
is the probability of a sequence of length n containing
x1 outcomes O1
x2 outcomes O2
…
xk outcomes Ok

n!

x1 ! x2 ! xk !  x1


xk 
n
x2
is the number of ways of choosing the positions for the x1
outcomes O1, x2 outcomes O2, …, xk outcomes Ok
 n  n  x1   n  x1  x2 

 

x
x
x
3
 1  2 

 xk 
 
 xk 




n  x1  !
n  x1  x2 !


n!

 x ! n  x  ! 
 x ! n  x  x ! 
 x ! n  x  x  x ! 
1
1
2
1
2
3
 1
 2
 3

n!

x1 ! x2 ! xk !
p  x1 ,
n!
, xn  
p1x1 p2x2
x1 ! x2 ! xk !


 x1
n
x2
pkxk
 x1 x2
 p1 p2
xk 
pkxk
is called the Multinomial distribution
Example:
Suppose that a treatment for back pain has three possible
outcomes:
O1 - Complete cure (no pain) – (30% chance)
O2 - Reduced pain – (50% chance)
O3 - No change – (20% chance)
Hence p1 = 0.30, p2 = 0.50, p3 = 0.20.
Suppose the treatment is applied to n = 4 patients suffering
back pain and let X = the number that result in a complete cure,
Y = the number that result in just reduced pain, and Z = the
number that result in no change.
Find the distribution of X, Y and Z. Compute P[X + Y ≥ Z]
4!
x
y
z
p  x, y , z  
 0.30   0.50   0.20 
x! y ! z !
x yz 4
Table: p(x,y,z)
x
0
0
0
0
0
1
1
1
1
1
2
2
2
2
2
3
3
3
3
3
4
4
4
4
4
y
0
1
2
3
4
0
1
2
3
4
0
1
2
3
4
0
1
2
3
4
0
1
2
3
4
0
0
0
0
0
0.0625
0
0
0
0.1500
0
0
0
0.1350
0
0
0
0.0540
0
0
0
0.0081
0
0
0
0
1
0
0
0
0.1000
0
0
0
0.1800
0
0
0
0.1080
0
0
0
0.0216
0
0
0
0
0
0
0
0
0
z
2
0
0
0.0600
0
0
0
0.0720
0
0
0
0.0216
0
0
0
0
0
0
0
0
0
0
0
0
0
0
3
0
0.0160
0
0
0
0.0096
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
4
0.0016
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
P [X + Y ≥ Z]
= 0.9728
x
0
0
0
0
0
1
1
1
1
1
2
2
2
2
2
3
3
3
3
3
4
4
4
4
4
y
0
1
2
3
4
0
1
2
3
4
0
1
2
3
4
0
1
2
3
4
0
1
2
3
4
0
0
0
0
0
0.0625
0
0
0
0.1500
0
0
0
0.1350
0
0
0
0.0540
0
0
0
0.0081
0
0
0
0
1
0
0
0
0.1000
0
0
0
0.1800
0
0
0
0.1080
0
0
0
0.0216
0
0
0
0
0
0
0
0
0
z
2
0
0
0.0600
0
0
0
0.0720
0
0
0
0.0216
0
0
0
0
0
0
0
0
0
0
0
0
0
0
3
0
0.0160
0
0
0
0.0096
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
4
0.0016
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Example: The Multivariate Normal distribution
Recall the univariate normal distribution
1

 
f  x 
e
2
 12
x
2
the bivariate normal distribution
f  x, y  
1
2 x y 1  
2
e
 12
2 1 





   2   
x x 2
x
x x
x
x y
y
  
x y 2 
y
The k-variate Normal distribution
f  x1 ,
, xk   f  x  
1
 2 
k /2

1/ 2
e
 12  x μ   1  x μ 
where
 x1 
x 
2

x
 
 
 xk 
 1 
 
μ   2
 
 
 k 
 11  12


12
22



 1k  2 k
 1k 
 2 k 


 kk 
Marginal distributions
Definition
Let X1, X2, …, Xq, Xq+1 …, Xk denote k discrete
random variables with joint probability function
p(x1, x2, …, xq, xq+1 …, xk )
then the marginal joint probability function
of X1, X2, …, Xq is
p12
q
 x , , x     p  x ,
1
q
1
xq1
xn
, xn 
Definition
Let X1, X2, …, Xq, Xq+1 …, Xk denote k continuous
random variables with joint probability density
function
f(x1, x2, …, xq, xq+1 …, xk )
then the marginal joint probability function
of X1, X2, …, Xq is

f12
q

 x , , x     f  x ,
1
q
1


, xn  dxq 1
dxn
Conditional distributions
Definition
Let X1, X2, …, Xq, Xq+1 …, Xk denote k discrete
random variables with joint probability function
p(x1, x2, …, xq, xq+1 …, xk )
then the conditional joint probability function
of X1, X2, …, Xq given Xq+1 = xq+1 , …, Xk = xk is
p1
q q 1 k
 x ,, x
1
q

xq 1 ,, xk 
p  x1 ,
pq 1
k
x
q 1
, xk 
,, xk 
Definition
Let X1, X2, …, Xq, Xq+1 …, Xk denote k continuous
random variables with joint probability density
function
f(x1, x2, …, xq, xq+1 …, xk )
then the conditional joint probability function
of X1, X2, …, Xq given Xq+1 = xq+1 , …, Xk = xk is
f1
q q 1 k
 x ,, x
1
q

xq 1 ,, xk 
f  x1 ,
f q 1
k
x
q 1
, xk 
,, xk 
Definition – Independence of sets of vectors
Let X1, X2, …, Xq, Xq+1 …, Xk denote k continuous
random variables with joint probability density
function
f(x1, x2, …, xq, xq+1 …, xk )
then the variables X1, X2, …, Xq are independent
of Xq+1, …, Xk if
f  x1 ,
, xk   f1
q
 x , , x  f
1
q
q 1 k
x
q 1
, , xk 
A similar definition for discrete random variables.
Definition – Mutual Independence
Let X1, X2, …, Xk denote k continuous random
variables with joint probability density function
f(x1, x2, …, xk )
then the variables X1, X2, …, Xk are called
mutually independent if
f  x1 ,
, xk   f1  x1  f 2  x2  f k  xk 
A similar definition for discrete random variables.
Example
Let X, Y, Z denote 3 jointly distributed random
variable with joint density function then


2

K
x
 yz 0  x  1, 0  y  1, 0  z  1

f  x, y , z   
0
otherwise


Find the value of K.
Determine the marginal distributions of X, Y and Z.
Determine the joint marginal distributions of
X, Y
X, Z
Y, Z
Solution
Determining the value of K.
  
1

  
1 1 1


f  x, y, z  dxdydz     K x 2  yz dxdydz
0 0 0
x 1
1 1
x

1

 K     xyz  dydz  K     yz  dydz
3
3

 x 0
0 0 
0 0
1 1
3
y 1
1
1
y 
1
1
 K   y  z  dz  K    z  dz
3
2  y 0
3
2
0
0
1
2
1
z z 
7
12
1 1
K   K  K
 1 if K 
12
7
3 4
 3 4 0
2
The marginal distribution of X.
f1  x  
 
1 1

 


12
2
f  x, y, z  dydz    x  yz dydz
7 00
y 1
1


12
y
12  2 1 
2
  x y 
z  dz    x  z  dz
7 0
2  y 0
7 0
2 
1
2
1
12  2
z  12  2 1 
  x z     x   for 0  x  1
7 
4 0 7 
4
2
The marginal distribution of X,Y.
f12  x, y  



1


12
2
f  x, y, z  dz   x  yz dz
7 0
z 1
12  2
z 
 x z  y 
7 
2  z 0
2
12  2 1 
  x  y  for 0  x  1, 0  y  1
7
2 
Find the conditional distribution of:
1.
Z given X = x, Y = y,
2.
Y given X = x, Z = z,
3.
X given Y = y, Z = z,
4. Y , Z given X = x,
5.
X , Z given Y = y
6.
X , Y given Z = z
7. Y given X = x,
8.
X given Y = y
9.
X given Z = z
10. Z given X = x,
11. Z given Y = y
12. Y given Z = z
The marginal distribution of X,Y.
12  2 1 
f12  x, y    x  y  for 0  x  1, 0  y  1
7
2 
Thus the conditional distribution of Z given X = x,Y = y
is
12 2
x  yz 

f  x, y , z 
 7
f12  x, y  12  2 1 
 x  y
7
2 
x 2  yz

1
2
x  y
2
for 0  z  1
The marginal distribution of X.
12  2 1 
f1  x    x   for 0  x  1
7
4
Thus the conditional distribution of Y , Z given X = x is
12 2
f  x, y, z  7  x  yz 

12  2 1 
f1  x 
x  
7
4
x 2  yz

for 0  y  1, 0  z  1
1
2
x 
4