Spring 2007 Geog 210C: Phaedon C. Kyriakidis Bivariate Distributions Definition: class of multivariate probability distributions describing joint variation of outcomes of two random variables (discrete or continuous), e.g., temperature and precipitation, household size and number of cars owned Example: survey results of 100 households reporting # of people per household (household size) and # of cars owned per household: # of cars owned (y) size (x) 0 1 2 3 tot. 2 10 8 3 2 23 3 7 10 6 3 26 4 4 5 12 6 27 5 1 2 6 15 24 tot. 22 25 27 26 100 Bivariate PMF: probability mass function f X,Y (x, y) that two RVs X and Y attain simultaneously two respective outcomes x and y: f X,Y (x, y)= P rob{X = x, Y = y} = number of samples occurring jointly in classes x and y / total number of samples # of cars owned (y) size (x) 0 1 2 3 f X (x) 2 0.10 0.08 0.03 0.02 0.23 3 0.07 0.10 0.06 0.03 0.26 4 0.04 0.05 0.12 0.06 0.27 5 0.01 0.02 0.06 0.15 0.24 f Y (y) 0.22 0.25 0.27 0.26 1.00 note that 0 ≤ f X,Y (x, y) ≤ 1, and ∑ x ∑ y f X,Y (x, y)=1 Slide 1 Discrete Bivariate Distribution Example # of cars owned (y) size (x) 0 1 2 3 f X (x) 2 0.10 0.08 0.03 0.02 0.23 3 0.07 0.10 0.06 0.03 0.26 4 0.04 0.05 0.12 0.06 0.27 5 0.01 0.02 0.06 0.15 0.24 f Y (y) 0.22 0.25 0.27 0.26 1.00 Stem plot representation: 2 2.5 3 3.5 4 4.5 5 0 1 2 3 0 0.05 0.1 0.15 0.2 household size Bivariate pmf example number of cars Slide 2 Lecture Notes Bivariate & Multivariate Distributions total # of slides = 22
11
Embed
Bivariate Distributionschris/Medrano_GEO 210C/GEO 210C... · Spring 2007 Geog 210C: Phaedon C. Kyriakidis Statistics of Indicators (2) n =10joint realizations of two RVs Z and Y:
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
Spring 2007 Geog 210C: Phaedon C. Kyriakidis
Bivariate Distributions
Definition: class of multivariate probability distributions describing
joint variation of outcomes of two random variables (discrete or
continuous), e.g., temperature and precipitation, household size and
number of cars owned
Example: survey results of 100 households reporting # of people per
household (household size) and # of cars owned per household:
# of cars owned (y)
size (x) 0 1 2 3 tot.
2 10 8 3 2 23
3 7 10 6 3 26
4 4 5 12 6 27
5 1 2 6 15 24
tot. 22 25 27 26 100
Bivariate PMF: probability mass function fX,Y (x, y) that two RVs
X and Y attain simultaneously two respective outcomes x and y:
fX,Y (x, y) = Prob{X = x, Y = y} = number of samples occurring
jointly in classes x and y / total number of samples
# of cars owned (y)
size (x) 0 1 2 3 fX (x)
2 0.10 0.08 0.03 0.02 0.23
3 0.07 0.10 0.06 0.03 0.26
4 0.04 0.05 0.12 0.06 0.27
5 0.01 0.02 0.06 0.15 0.24
fY (y) 0.22 0.25 0.27 0.26 1.00
note that 0 ≤ fX,Y (x, y) ≤ 1, and∑
x
∑y fX,Y (x, y) = 1
Slide 1
Discrete Bivariate Distribution Example
# of cars owned (y)
size (x) 0 1 2 3 fX (x)
2 0.10 0.08 0.03 0.02 0.23
3 0.07 0.10 0.06 0.03 0.26
4 0.04 0.05 0.12 0.06 0.27
5 0.01 0.02 0.06 0.15 0.24
fY (y) 0.22 0.25 0.27 0.26 1.00
Stem plot representation:
22.5
33.5
44.5
5
0
1
2
30
0.05
0.1
0.15
0.2
household size
Bivariate pmf example
number of cars
Slide 2
Lecture Notes Bivariate & Multivariate Distributions total # of slides = 22
Spring 2007 Geog 210C: Phaedon C. Kyriakidis
Discrete Marginal Distributions
# of cars owned (y)
size (x) 0 1 2 3 fX (x)
2 0.10 0.08 0.03 0.02 0.23
3 0.07 0.10 0.06 0.03 0.26
4 0.04 0.05 0.12 0.06 0.27
5 0.01 0.02 0.06 0.15 0.24
fY (y) 0.22 0.25 0.27 0.26 1.00
Marginal (univariate) distributions:
• univariate PMFs of the two RVs X and Y
• obtained by integrating (summing up) the columns and rows of
the PMF table:
fX(x) =∑
y
fX,Y (x, y) and fY (y) =∑
x
fX,Y (x, y)
fX(x) = Prob{X = x, Y ∈ [−∞, +∞]}marginal PMF = bivariate PMF with one RV unconstrained
Examples:
• probability of a randomly selected household with 3 members:
fX(3) = 0.07 + 0.10 + 0.06 + 0.03 = 0.26
• probability of a randomly selected household with no cars:
fY (0) = 0.10 + 0.07 + 0.04 + 0.01 = 0.22
Slide 3
Discrete Conditional Distributions
# of cars owned (y)
size (x) 0 1 2 3 fX (x)
2 0.10 0.08 0.03 0.02 0.23
3 0.07 0.10 0.06 0.03 0.26
4 0.04 0.05 0.12 0.06 0.27
5 0.01 0.02 0.06 0.15 0.24
fY (y) 0.22 0.25 0.27 0.26 1.00
Conditional distribution:
• PMF of one RV, say Y , given an outcome, X = x of another
RV, say X:
fY |X(y|x) = Prob{Y = y|X = x}
• obtained by dividing (standardizing) the PMF fY,X(y, x) by the
marginal PMF fX(x) of conditioning event:
fY |X(y|x) =fY,X(y, x)
fX(x)=
fX,Y (x, y)
fX(x)
division by fX(x) standardizes (to unit sum) PMF entries corresponding
to particular row (column):∑
y
fY |X(y|x) = 1
Examples:
• conditional PMF of car ownership, given 3-member households:
fY |X (0|3) =0.07
0.26fY |X (1|3) =
0.10
0.26fY |X (3|3) =
0.03
0.26
• conditional PMF of household size, given 3-car ownership:
fX|Y (2|3) =0.02
0.26fX|Y (3|3) =
0.03
0.26fX|Y (4|3) =
0.06
0.26
Slide 4
Lecture Notes Bivariate & Multivariate Distributions total # of slides = 22