Download Marginal and conditional densities Let a random point (X, Y ) be

Probability & Statistics
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Marginal and conditional densities
Let a random point (X, Y ) be distributed in the triangular region ∆ bounded by
x = 0, y = 0, and x + 2y = 2. Suppose that the density f = fX,Y of (X, Y ) is
(x, y) ∈ ∆.
f (x, y) = 3y,
In other words, the chance of finding (X, Y ) in a tiny rectangle [x, x + dx] × [y, y + dy],
positioned entirely within ∆, is f (x, y)dxdy = 3y dxdy. The total probability mass is
∫∫
3y dxdy = 1.
∆
Note that, although the upper y-layers of ∆ are shorter than lower y-layers, they have a
higher concentration of probability.
The marginal density of X is obtained by pushing all of the mass down onto the x-axis.
∫
fX (x) = f (x, y)dy
y
∫
1− 21 x
3ydy
=
0
3
=
2
(
1
1− x
2
)2
Here is the plot of the marginal density fX (x) :
∫
Check that
2
fX (x)dx = 1.
0
,
0 < x < 2.
The marginal density of Y is obtained by compressing the y-layers to the y-axis.
∫
fY (y) = f (x, y)dx
x
∫ 2−2y
=
3ydx
0
= 6y(1 − y),
0 < y < 1.
Here is the plot of the marginal density fY (y) :
∫
Check that
1
fY (y)dy = 1.
0
We may also calculate various conditional densities. For instance,
f (1, y)
fY |X (y|1) =
fX (1)
3y
=
3/8
= 8y,
0 < y < 1/2.
∫ 1/2
Check that
fY |X (y|1)dy = 1. At the same time,
0
f (x, 1/2)
fY (1/2)
3/2
=
3/2
= 1,
0 < x < 1.
fX|Y (x|1/2) =