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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.
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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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(or a two dimensional random vector).
r.v.
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.
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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
≤ ≤
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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
= (^) =
= =
∑∑
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.
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two dimensional random vector (x, y) is defined as
F(x, y) = P[X<x, Y<y]
It follows from the definition, that
( , )
y (^) x
f x y dxdy
−∞ −∞
=
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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
Example 1
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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
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Example 1 (contd)
2 2
1 9 5
1 5 9 5 5
25 2 2
1 0.
⎧ (^) ⎡ ⎤⎫ ⎪ ⎪
= − − × − + × ⎨ ⎬ ⎢ ⎥
⎪ (^) ⎣ ⎦⎪ ⎩ ⎭
= −
=
P[X > Y] = 0.
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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
∫ ∫
∫
∫
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Example 2 (contd)
3
2
(^0 )
3
2
0
3 3 3 3
0
y x
x
x
∫
∫
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
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Limits x → 0 to y
y → 0 to 1
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
y
x > y
x = y
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Limits x → 0 to 1
y → 1-x to 1
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
−
∫ ∫
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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
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Y = y j
] and P[X = x i
] resply.
by marginal sum as 0.25.
X=5. we have P[Y=0, X=0 OR Y=0, X=1 OR …..]
This indicates P[Y=0] irrespective of the value of X
