Page 66 - Elements of Distribution Theory

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52 Conditional Distributions and Expectation

Hence, the conditional distribution of X given Y = y is absolutely continuous with density
function
1 − x − y
p X|Y (x|y) = 2 , 0 < x < 1 − y
(1 − y) 2
where 0 < y < 1.

Example 2.12 (Trinomial distribution). Let (X 1 , X 2 , X 3 )bea random vector with a
multinomial distribution with parameters n and (θ 1 ,θ 2 ,θ 3 ); see Example 2.2. This is some-
times called a trinomial distribution. It was shown in Example 2.2 that the marginal distri-
bution of X 1 is binomial with parameters n and θ 1 .
Hence, (X 1 , X 2 , X 3 ) has frequency function

n
x 1 x 2 x 3
p(x 1 , x 2 ) = θ θ θ
1 2 3
x 1 , x 2 , x 3
and the marginal frequency function of X 1 is given by
n

x 1
(x 1 ) = θ (1 − θ 1 ) n−x 1 .
1
p X 1
x 1
It follows that the conditional distribution of (X 2 , X 3 )given X 1 = x 1 is discrete with fre-
quency function
n x 1 x 2 x 3 x 2 x 3 )
θ θ θ θ θ
x 1 ,x 2 1 2 3 n − x 1 2 3
= ,
n x 1
p X 2 ,X 3 |X 1 (x 2 , x 3 |x 1 ) =
θ (1 − θ 1 ) n−x 1 x 2 , x 3 (θ 2 + θ 3 ) n−x 1
x 1 1
where x 2 , x 3 = 0,..., n − x 1 with x 2 + x 3 = n − x 1 , for x 1 = 0,..., n; recall that θ 1 +
θ 2 + θ 3 = 1.
That is, the conditional distribution of (X 2 , X 3 )given X 1 = x 1 is multinomial
with parameters n − x 1 and (θ 2 /(θ 2 + θ 3 ),θ 3 /(θ 2 + θ 3 )). Alternatively, we can say
X 2 has a binomial distribution with parameters n − x 1 and θ 2 /(θ 2 + θ 3 ) with X 3 =
n − x 1 − X 2 .
Example 2.13 (A mixed distribution). Let (X, Y) denote a two-dimensional random vec-
tor with the distribution described in Example 2.3. Recall that this distribution is neither
absolutely continuous nor discrete.
First consider the conditional distribution of X given Y = y. Recall that for A ⊂ R and
+
y = 1, 2,
1
Pr(X ∈ A, Y = y) = y exp(−yx) dx.
2 A
Note that
1
Pr(X ≤ x, Y = y) = [1 − exp(−yx)], x > 0, y = 1, 2
2
so that, for y = 1, 2,

Pr(X ≤ x|Y = y) = 1 − exp(−yx), x > 0.