Download Notes 17 - Wharton Statistics

Statistics 510: Notes 17
Reading: Sections 6.1-6.2
I. Joint distribution functions (Chapter 6.1)
Thus far, we have focused on probability distributions for
single random variables. However, we are often interested
in probability statements concerning two or more random
variables. The following examples are illustrative:
 In ecological studies, counts, modeled as random
variables, of several species are often made. One
species is often the prey of another; clearly, the
number of predators will be related to the number of
prey.
 The joint probability distribution of the x, y and z
components of wind velocity can be experimentally
measured in studies of atmospheric turbulence.
 The joint distribution of the values of various
physiological variables in a population of patients is
often of interest in medical studies.
 A model for the joint distribution of age and length in
a population of fish can be used to estimate the age
distribution from the length distribution. The age
distribution is relevant to the setting of reasonable
harvesting policies.
The joint behavior of two random variables X and Y is
determined by the joint cumulative distribution function
(cdf):
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FXY ( x, y)  P( X  x, Y  y) ,
where X and Y are continuous or discrete.
The probability that ( X , Y ) belongs to a given rectangle is
P( x1  X  x2 , y1  Y  y2 ) 
F (x , y )  F (x , y )  F (x , y )  F (x , y ) .
2
2
2
1
1
2
1
1
In general, if X 1 , , X n are jointly distributed random
variables, the joint cdf is
F ( x1 , , xn )  P( X1  x1 , X 2  x2 , , X n  xn ) .
Two- and higher-dimensional versions of probability
distribution functions and probability mass functions exist.
We start with a detailed description of joint probability
mass functions.
Joint probability mass functions: Let X and Y be discrete
random variables defined on the sample space that take on
values x1 , x2 , and y1 , y2 , respectively. The joint
probability mass function of ( X , Y ) is
p( xi , y j )  P( X  xi , Y  y j ) .
Example 1: A fair coin is tossed three times independently:
let X denote the number of heads on the first toss and Y
denote the total number of heads. Find the joint probability
mass function of X and Y.
The joint pmf is given in the following table:
2
y
0
1/8
0
x
0
1
1
2/8
1/8
2
1/8
2/8
3
0
1/8
Marginal probability mass functions: Suppose that we wish
to find the pmf of Y from the joint pmf of X and Y.
pY (0)  P (Y  0)
 P (Y  0, X  0)  P (Y  0, X  1)
1
 0
8
1

8
pY (1)  P (Y  1)
 P (Y  1, X  0)  P (Y  1, X  1)
2 1
 
8 8
3

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In general, to find the frequency function of Y, we simply
sum down the appropriate column of the table giving the
joint pmf of X and Y. For this reason, pY is called the
marginal probability mass function of Y. Similarly,
summing across the rows gives
p X ( x)   p( x, yi )
i
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which is the marginal pmf of X.
The case for several random variables is analogous. If
X 1 , , X n are discrete random variables defined on the
sample space, their joint pmf is
p( x1 , , xn )  P( X1  x1 , , X n  xn ) .
The marginal pmf of X1 , for example, is
pX1 ( x1 ) 

p( x1 , x2 ,
x2 , , xn
, xn ) .
Example 2: One of the most important joint distributions is
the multinomial distribution which arises when a sequence
of n independent and identical experiments is performed,
where each experiment can result in any one of r possible
outcomes, with respective probabilities p1 , , pr ,
p1 , , pr . If we let X i denote the number of the n
experiments that result in outcome i, then
n!
P( X 1  n1 , , X r  nr ) 
p1n1 prnr
n1 ! nr !
r
whenever
n
i 1
i
 n.
Suppose that a fair die is rolled 9 times. What is the
probability that 1 appears three times, 2 and 3 twice each, 4
and 5 once each and 6 not at all?
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II. Jointly continuous random variables
Joint PDF and Joint CDF: Suppose that X and Y are
continuous random variables. The joint probability density
function (pdf) of X and Y is the function f ( x, y ) such that
for every set C of pairs of real numbers
P(( X , Y )  C )   f ( x, y )dxdy
( x , y )C
Another interpretation of the joint pdf is obtained as
follows:
P{a  X  a  da, b  Y  b  db}  
b  db
b

a  da
a
f ( x, y )dxdy
 f (a, b)dadb
when da and db are small and f ( x, y ) is continuous at a , b .
Hence f (a, b) is a measure of how likely it is that the
random vector ( X , Y ) will be near (a, b) This is similar to
the interpretation of the pdf f ( x ) for a single random
variable X being a measure of how likely it is to be near x .
The joint CDF of ( X , Y ) can be obtained as follows:
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F (a, b)  P{ X  ( , a ], Y  ( , b]}


f ( x, y )dxdy
X (  , a ],Y (  ,b ]

b

a
 
f ( x, y )dxdy
It follows upon differentiation that
2
f ( a, b) 
F ( a, b)
ab
wherever the partial derivatives are defined.
Marginal PDF: The cdf and pdf of X can be obtained from
the pdf of ( X , Y ) :
P( X  x)  P{ X  x, Y  (, )}

x


 
f ( x, y)dydx

d
d x 
P( X  x) 
f
(
x
,
y
)
dydx

 f ( x, y)dy
dx
dx  
Similarly, the pdf of Y is given by
f X ( x) 
fY ( y)  


f ( x, y)dx
Example 3: Consider the pdf for X and Y
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f ( x, y )  ( x 2  xy ), 0  x  1, 0  y  1
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(a) Find P ( X  Y )
(b) Find the marginal density f X ( x) of X .
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III. Independent Random Variables (Chapter 6.2)
The random variables X and Y are said to be independent
if for any two sets of real numbers A and B ,
P{ X  A, Y  B}  P{ X  A}P{Y  B} .
Loosely, speaking X and Y are independent if knowing the
value of one of the random variables does not change the
distribution of the other random variable (we will develop
this interpretation more in Chapter 6.4-6.5). Random
variables that are not independent are said to be dependent.
For discrete random variables, the condition of
independence is equivalent to
p( x, y)  pX ( x) pY ( y) for all x, y .
For continuous random variables, the condition of
independence is equivalent to
f ( x, y)  f X ( x) fY ( y) for all x, y .
Example 4: Consider ( X , Y ) with joint pmf
1
p (10,1)  p(20,1)  p(20, 2) 
10
1
3
p (10, 2)  p(10,3)  , and p(20,3) 
5
10
Are X and Y independent?
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Example 5: Suppose that a node in a communications
network has the property that if two packets of information
arrive within time  of each other they “collide” and then
have to be retransmitted. If the times of arrival of the two
packets are independent and uniform on [0, T ] , what is the
probability that they collide?
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