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Probability Distributions of 2D Random Variables: Joint, Marginal, and Conditional, Study notes of Mathematical Statistics

An in-depth analysis of probability distributions for two-dimensional random variables, including joint, marginal, and conditional distributions. It covers various examples and formulas for obtaining marginal and conditional distributions from joint distributions. The document also discusses the concept of independent random variables.

Typology: Study notes

2012/2013

Uploaded on 04/20/2013

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Bivariate%DistribuAons%
In many situations, we would be interested in
simultaneous behavior of two or more random variables.
e.g., in hydrology, we may be interested in the joint
behavior of
ØRainfall – Runoff
ØRainfall – Recharge
ØRainfall intensity- Peak flood discharge
ØTemperature – Evaporation
ØSoil permeability – GW yield
ØFlow rates on two streams
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5

Bivariate DistribuAons

  • In many situations, we would be interested in

simultaneous behavior of two or more random variables.

e.g., in hydrology, we may be interested in the joint

behavior of

Ø Rainfall – Runoff

Ø Rainfall – Recharge

Ø Rainfall intensity- Peak flood discharge

Ø Temperature – Evaporation

Ø Soil permeability – GW yield

Ø Flow rates on two streams

6

  • We denote (X,Y) as a two-dimensional random variable

(or a two dimensional random vector).

  • X and Y both discrete : two dimensional discrete r.v
  • X and Y both continuous : two dimensional continuous

r.v.

  • It is possible that one of the rvs of (X, Y), say, X, is

discrete while the other is continuous. In this course,

however, we deal only with cases in which both X & Y

are either discrete or continuous.

Bi-­‐variate DistribuAons

8

  • Similar to the CDF for one dimensional random

variables, we define the joint CDF of the two

dimensional discrete r.v., (X,Y) as

F(x, y) = Prob [X<x, Y<y]

i j

i j

x x y y

p x y

≤ ≤

Probability DistribuAon of (X, Y)

F(∞, ∞) = P[X< ∞, Y< ∞] = 1.

9

Example : Discrete two-­‐d RV

F(3,2) =

P[X<3, Y<2] = 0+0.04+0.05+0.07+0.03+0.04+0.06+0.07+

0.02+.0.05+.0.05+0.07+0.01+0.03+0.05+0.

= 0.

(^2 )

0 0

y (^) x

y x

p x y

= (^) =

= =

∑∑

Probability mass function

0 0.04 0.

0.05 0.05 0.

0.06 0.

0.03 0.

x

y

0 1 2 3 4

0 0.

1 0.

2 0.

3 0.01 0.03 0.05 0.07 0.

11

  • The joint cumulative distribution function F(x, y) of the

two dimensional random vector (x, y) is defined as

F(x, y) = P[X<x, Y<y]

It follows from the definition, that

• F(∞, ∞)=1.
  • F(- ∞, y) = F(x, - ∞) = 0

Joint cdf of (X, Y)

( , )

y (^) x

f x y dxdy

−∞ −∞

=

12

Flows in two adjacent streams are denoted as a random

vector (X, Y) with a joint pdf

f(x, y) = c if 5 < x < 10 ; 4 < y < 9

= 0, elsewhere

  1. Obtain c
  2. Obtain P[X > Y]

Example 1

X
Y

14

2. P[X > Y] = 1-P[X < Y]

Example 1 (contd)

9

5 5

9

5 5

9

5

9 2

5

1 ( , )

1

1

25

1

1 ( 5)

25

1

1 5

25 2

y

y

f x y dxdy

dxdy

y dy

y

y

= −

= −

⎧ ⎫ ⎪ ⎪

= − − ⎨ ⎬

⎪ ⎪ ⎩ ⎭

⎧ ⎫

⎪ ⎡^ ⎤ ⎪

= − − ⎨ ⎬ ⎢ ⎥

⎣ ⎦ ⎪ ⎪ ⎩ ⎭

∫ ∫

∫ ∫

x

y

x ≤ y

15

Example 1 (contd)

2 2

1 9 5

1 5 9 5 5

25 2 2

1 0.

⎧ (^) ⎡ ⎤⎫ ⎪ ⎪

= − − × − + × ⎨ ⎬ ⎢ ⎥

⎪ (^) ⎣ ⎦⎪ ⎩ ⎭

= −

=

P[X > Y] = 0.

17

  1. To obtain c ,

Example 2 (contd)

f ( , x y dxdy ) 1

∞ ∞

−∞ −∞

=

∫ ∫

1 1

2 2

0 0

1 (^1 )

2

(^0 )

1 2 0 1 3 0

c x y dxdy

x

c xy dy

c y dy

y y c

c c

∫ ∫

18

  1. F(x, y) =

Example 2 (contd)

3

2

(^0 )

3

2

0

3 3 3 3

0

y x

x

x

y

x y dx

y

x y dx

x y xy x y xy

2 2

0 0 0 0

3

( , )

2

x y^ x y

f x y dydx = x + y dydx

∫ ∫ ∫ ∫

3 3

( , )

2

x y xy

F x y

=

0 < x < 1

0 < y < 1

20

4. P[Y > X]

Limits x → 0 to y

y → 0 to 1

P[Y > X] =

Example 2 (contd)

1

2 2

0 0

3

2

y

x + y dxdy

∫ ∫

1 (^1 3 2 )

3

0 0

1 3 4

0

y

x xy y

dy y dy

y y y

∫ ∫

x

y

x > y

x = y

21

5. P[X+Y > 1]

Limits x → 0 to 1

y → 1-x to 1

P[X+Y> 1] =

Example problem-­‐

1 1

2 2

0 1

3

2 x

x y dydx

∫ ∫

x

y

1 (^1 2 3 )

3

0 1 0

1 2 3 4

0

x

x y y x x

dx x dx

x x y

∫ ∫

23

Marginal Probability DistribuAon

e.g., P[X < 3] = 0.06+0.16+0.21+0.28 = 0.

x

y

0 1 2 3 4 Sum

(^0 0) 0.04 0.05 0.07 0.09 0.

1 0.03 0.04 0.06 0.07 0.08 0.

2 0.02 0.05 0.05 0.07 0.05 0.

(^3) 0.01 0.03 0.05 0.07 0.07 0.

Sum 0.06 0.16 0.21 0.28 0.29 1.

Marginal distribution of X

Marginal distribution of Y

24

Marginal Probability DistribuAon

  • An element in the body of the table indicates P[X = x i

Y = y j

].
  • The marginal totals give P[Y = y j

] and P[X = x i

] resply.

  • For example, if we are interested in P[Y = 0], this is given

by marginal sum as 0.25.

  • Since the event P[Y = 0] can occur with X=0, X=1,……..

X=5. we have P[Y=0, X=0 OR Y=0, X=1 OR …..]

P[Y = 0] = P[Y=0, X=0]+P[Y=0, X=1]+ P[Y=0, X=2]+……
……………. P[Y=0, X=5]

This indicates P[Y=0] irrespective of the value of X