Transcript C9_Math3033

MATH 3033 based on
Dekking et al. A Modern Introduction to Probability and Statistics. 2007
Slides by Michael Maurizi
Format by Tim Birbeck
Instructor Longin Jan Latecki
C9: Joint Distributions and Independence
9.1 – Joint Distributions of Discrete
Random Variables

Joint Distribution: the combined distribution of two or more random
variables defined on the same sample space Ω

Joint Distribution of two discrete random variables:
The joint distribution of two discrete random variables X and Y can be
obtained by using the probabilities of all possible values of the pair (X,Y)
Joint Probability Mass function p of two discrete random variables X and Y:
p : R 2  [0,1]
p XY (a, b)  p(a, b)  P(X  a, Y  b)
for    a, b  
Joint Distribution function F of two random variables X and Y: Can be thought
of as the sum of the elements in box it makes with the upper-left corner.
p : R 2  [0,1]
F (a, b)  P(X  a, Y  b)
for    a, b  
9.1 – Joint Distributions of Discrete
Random Variables

Marginal Distribution:
Obtained by adding up the
rows or columns of a joint
probability mass function table.
Literally written in the margins.
Example: Joint Distribution of S and M.
S = The sum of two dice, M = The maximum of two dice.
b
pS(a)
a
1
2
3
4
5
6
2
1/36
0
0
0
0
0
1/36
3
0
2/36
0
0
0
0
2/36
4
0
1/36
2/36
0
0
0
3/36
5
0
0
2/36
2/36
0
0
4/36
6
0
0
1/36
2/36
2/36
0
5/36
7
0
0
0
2/36
2/36
2/36
6/36
Marginal distribution function of Y:
8
0
0
0
1/36
2/36
2/36
5/36
FY (b)  P(Y  b)  F (b,)  lim F (a, b)
9
0
0
0
0
2/36
2/36
4/36
10
0
0
0
0
1/36
2/36
3/36
11
0
0
0
0
0
2/36
2/36
12
0
0
0
0
0
1/36
1/36
pM(a)
1/36
3/36
5/36
7/36
9/36
11/36
1
Marginal distribution function of X:
FX (a)  P(X  a)  F (a,)  lim F (a, b)
b
a
9.2 – Joint Distributions of Continuous
Random Variables

Joint Continuous Distribution: Like an ordinary continuous random
variable, only works for a range of values. There must exist a function f
that fulfills the following properties for there to be a joint continuous
distribution:
2
f :R  R
b1 b2
P(a1  X  b1 , a2  Y  b2 )    f ( x, y )dxdy
for all a1  b1 and a2  b2
a1 a2
f ( x, y )  0
 
for all x and y
  f ( x, y)dxdy  1
  
9.2 – Joint Distributions of Continuous
Random Variables
Joint distribution function:
F(a,b) can be constructed given f(x,y), and vice versa
a b
F ( a, b) 
  f ( x, y)dxdy
and
  
2
f(x,y) 
F ( x, y )
xy
Marginal probability density function:
You need to integrate out the unwanted random variable to get the
marginal distribution.

f X ( x) 
 f ( x, y)dy


and
fY ( y ) 
 f ( x, y)dx

9.3 – More than Two Random Variables
Assuming we have n random variables X1, X2, X3, … Xn. We can get the
joint distribution function and the joint probability mass functions.
for    a1 , a2 , , an  
Joint distributi on function :
F (a1 , a2 , , an )  P(X1  a1 , X 2  a2 ,, X n  an )
Joint probabilit y mass function :
p(a1 , a2 , , an )  P(X1  a1 , X 2  a2 ,, X n  an )
9.4 – Independent Random Variables
Tests for Independence: Two random variables X and Y are independent if
and only if every event involving X is independent of every event involving Y.
This also applies to joint distributions using more than two random variables.
for all possible a and b
P(X  a, Y  b)  P(X  a )  P(Y  b)
or
P(X  a, Y  b)  P(X  a )  P(Y  b)
or
F (a, b)  FX (a )  FY (b)
9.5 – Propagation of Independence
Independence after a change of variable:
If a function is applied to several independent random variables, the new
resulting random variables will also be independent.