Lecture 10: Joint Distributions & Order Statistics
Lecture 10: Joint Distributions & Order Statistics
Sta 111
Colin Rundel
May 28, 2014
Joint Distributions Discrete RVs
Joint Distribution - Example, cont.
Let B be the number of Black socks and W the number of White socks drawn, then the joint distribution of B and W is given by:
W
012
0
1 8 6 15 66 66 66 66
B
1
12 66
24 66
0
36 66
2
15 66
0
0
15 66
28 32 6 66 66 66 66 66
1/66 8/66 6/66 12/66 24/66 P(B = b, W = w ) = 0/66 15/66 0/66 0/66 0
If b=0,w=0 If b=0,w=1 If b=0,w=2 If b=1,w=0 If b=1,w=1 If b=1,w=2 If b=2,w=0 If b=2,w=1 If b=2,w=2 otherwise
64
2
P(B = b, W = w ) =
b
w
2-b-w 12
, for 0 b, w 2 and b + w 2
2
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Lecture 10
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Joint Distributions Discrete RVs
Joint Distribution - Example
Draw two socks at random, without replacement, from a drawer full of twelve colored socks:
6 black, 4 white, 2 purple
Let B be the number of Black socks, W the number of White socks drawn, then the distributions of B and W are given by:
0
1
2
P(B=k)
6 12
5 11
=
15 66
2
6 12
6 11
=
36 66
6 12
5 11
=
15 66
P(W=k)
8 12
7 11
=
28 66
2
4 12
8 11
=
32 66
4 12
3 11
=
6 66
Note - B HyperGeo(12, 6, 2) =
Sta 111 (Colin Rundel)
66 k 2-k
12 2
and W HyperGeo(12, 4, 2) =
Lecture 10
48 k 2-k
12 2
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Joint Distributions Discrete RVs
Marginal Distributions
Note that the column and row sums are the distributions of B and W respectively.
P(B = b) = P(B = b, W = 0) + P(B = b, W = 1) + P(B = b, W = 2) P(W = w ) = P(B = 0, W = w ) + P(B = 1, W = w ) + P(B = 2, W = w )
These are the marginal distributions of B and W . In general,
P(X = x) = P(X = x, Y = y ) = P(X = x|Y = y )P(Y = y )
y
y
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Lecture 10
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Joint Distributions Discrete RVs
Conditional Distribution
Conditional distributions are defined as we have seen previously with
P(X = x|Y = y ) = P(X = x, Y = y ) = joint pmf
P(Y = y )
marginal pmf
Therefore the pmf for white socks given no black socks were drawn is
1
P (W
=
w |B
=
0)
=
P(W = w , B = P(B = 0)
0)
=
66
8
66
6
66
15 66
=
1 15
15 66
=
8 15
15 66
=
6 15
if W = 0 if W = 1 if W = 2
Joint CDF
Joint Distributions Continuous RVs
F (x, y ) = P[X x, Y y ] = P[(X , Y ) lies south-west of the point (x, y )]
Y (x,y)
q
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Lecture 10
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Joint CDF, cont.
Joint Distributions Continuous RVs
The joint Cumulative distribution function follows the same rules as the univariate CDF,
Univariate definition:
lim F (x) = 0
x -
x
F (x) = P(X x) =
f (z)dz
-
lim F (x) = 1
x
x y F (x) F (y )
Bivariate definition:
y
x
F (x, y ) = P(X x, Y y ) =
f (x, y ) dx dy
- -
lim F (x, y ) = 0
x,y - Sta 111 (Colin Rundel)
lim F (x, y ) = 1
x,y Lecture 10
x x ,y y F (x, y ) F (x , y )
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Sta 111 (Colin Rundel)
X
Lecture 10
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Marginal CDFs
Joint Distributions Continuous RVs
We can define marginal CDFs using the joint CDF by setting one of the values to infinity:
x
F (x, ) = P(X x, Y ) =
f (x, y ) dy dy
- -
= P(X x) = FX (x)
y
F (, y ) = P(X , Y y ) =
f (x, y ) dx dy
- -
= P(Y y ) = FY (y )
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Joint pdf
Joint Distributions Continuous RVs
Similar to the CDF the probability density function follows the same general rules in two dimensions,
Univariate definition:
f (x) 0 for all x
f (x)
=
d dx
F
(x
)
-
f
(x )dx
=
1
Bivariate definition:
f (x, y ) 0 for all (x, y )
f (x, y ) =
F (x, y )
x y
f (x, y ) dx dy = 1
- -
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Lecture 10
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Joint Distributions Continuous RVs
Probability and Expectation
Univariate definition:
P(X A) = f (x) dx
A
E [g (X )] =
g (x) ? f (x) dx
-
Bivariate definition:
P(X A, Y B) =
f (x, y ) dx dy
AB
E [g (X , Y )] =
g (x, y ) ? f (x, y ) dx dy
- -
Marginal pdfs
Joint Distributions Continuous RVs
Marginal pdfs are derived by integrating out one of the random variables.
fX (x) = fY (x) =
f (x, y ) dy
-
f (x, y ) dx
-
Previously we defined independence in terms of, X and Y are independent if and only if E (XY ) = E (X )E (Y ).
An equivalent definitions is, X and Y are independent if and only if f (x, y ) = fX (x)fY (y ).
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Joint Distributions Continuous RVs
Example 1 - Joint Uniforms
Let f (x, y ) = c for x (0, 1), y (0, 1). Find c.
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Example 1, cont
Joint Distributions Continuous RVs
Given the f (x, y ) we just found, find F (x, y ).
Example 1, cont.
Joint Distributions Continuous RVs
Verify that the F (x, y ) we just found gives the correct f (x, y ),
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Example 1, cont.
Joint Distributions Continuous RVs
Check that f (x, y ) produces the correct marginal densities for X and Y (fX (x) and fY (y ))
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Lecture 10
Example 1, cont.
Joint Distributions Continuous RVs
Find the expected value of X and Y
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Example 1, cont.
Joint Distributions Continuous RVs
Find the expected value of XY
Example 2
Joint Distributions Continuous RVs
Let f (x, y ) = cx2y for x2 y 1.
Find: a) c b) P(|X | Y ) c) fX (x) and fY (y )
Sta 111 (Colin Rundel)
Lecture 10
Joint Distributions Continuous RVs
Example 2 - Range
1
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Sta 111 (Colin Rundel)
Lecture 10
Example 2.a
Find c
Joint Distributions Continuous RVs
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Y
-1 Sta 111 (Colin Rundel)
0 X
Lecture 10
1 May 28, 2014 18 / 40
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Example 2 - pdf
Joint Distributions Continuous RVs
0.15
f(x,y) 0.10
0.05
0.00
1.0
0.8
1.0
0.6
0.5
0.4
y
0.2
0.0 -1.0
-0.5
0.0 x
Sta 111 (Colin Rundel)
Lecture 10
Joint Distributions Continuous RVs
Example 2.b, cont.
Example 2.b
Joint Distributions Continuous RVs
0.20
Find P(|X | Y ).
To do this we need to integrate over the region where x2 y 1 and
0.15
|x| y which is indicated in red below
0.10
1
1
0.05
Y
Y
0.00
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-1
0
X
Sta 111 (Colin Rundel)
1
-1
Lecture 10
Example 2.c
Joint Distributions Continuous RVs
Find the marginal densities
0 X
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1
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Joint Distributions Continuous RVs
Example 2.c, cont.
It is always a good idea to check that the marginals are proper densities.
Example 3
Joint Distributions Continuous RVs
Let Y be the rate of calls at a help desk, and X the number of calls between 2 pm and 4 pm one day; Let's say that:
f (x, y ) = (2y )x e-3y x!
for y > 0, x = 0, 1, 2, . . ..
Find: a) P(X = 0) b) P(Y > 2)
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Lecture 10
Example 3.a
Find P(X = 0)
Joint Distributions Continuous RVs
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Lecture 10
Example 3.b
Find P(Y > 2)
Joint Distributions Continuous RVs
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Order Statistics
Order Statistics
Let X1, X2, X3, X4, X5 be iid random variables with a distribution f on the range (a, b). We can relabel these X's such that their labels correspond to arranging them in increasing order
X(1) X(2) X(3) X(4) X(5).
X(1)
a
X5
X(2)
X(3)
X(4)
X1
X4
X2
X(5)
X3
b
In the case where the distribution f is continuous we can make the stronger statement
X(1) < X(2) < X(3) < X(4) < X(5). Since P(Xi = Xj ) = 0 for all i = j for continuous random variables.
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Notational Detour
Order Statistics
For a continuous random variable
f (x) P(x X x + ) = P(X [x, x + ]) f (x) P(x X x + ) = P(X [x, x + ])/
f (x) = lim P(X [x, x + ])/ 0
P (x X x + ) = P (X 2 [x, x + ])
Sta 111 (Colin Rundel)
f (x) f (x + )
x x+ Lecture 10
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Order Statistics
Order Statistics, cont.
For X1, X2, . . . , Xn iid random variables X(k) is the kth smallest X , usually called the kth order statistic. The first order statistic, X(1) is therefore the smallest Xi and
X(1) = min(X1, . . . , Xn)
Similarly, the nth order statistic, X(n) is the largest Xi and X(n) = max(X1, . . . , Xn)
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Order Statistics
Density of the maximum
For X1, X2, . . . , Xn iid continuous random variables with pdf f and cdf F the density of the maximum is
P(X(n) [x, x + ]) = P(one of the X 's [x, x + ] and all others < x)
n
= P(Xi [x, x + ] and all others < x)
i =1
= nP(X1 [x, x + ] and all others < x) = nP(X1 [x, x + ])P(all others < x) = nP(X1 [x, x + ])P(X2 < x) ? ? ? P(Xn < x) = nf (x) F (x)n-1
f(n)(x ) = nf (x )F (x )n-1
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