PDF of Lecture Notes - School of Mathematical Sciences
PDF of Lecture Notes - School of Mathematical Sciences
PDF of Lecture Notes - School of Mathematical Sciences
You also want an ePaper? Increase the reach of your titles
YUMPU automatically turns print PDFs into web optimized ePapers that Google loves.
1. DISTRIBUTION THEORY<br />
Pro<strong>of</strong>.<br />
M X (t) = E[e tY ]<br />
= E[e t(X 1+X 2 ,···+X r) ]<br />
= E[e tX 1+tX 2 +···+tX r<br />
]<br />
= M X (t, t, t, . . . , t) (using def 1.9.3).<br />
For parts (2) and (3), substitute into (1) and use Theorem 1.9.6.<br />
Examples<br />
Consider RVs X, V , defined by V ∼ Gamma(α, λ) and the conditional distribution <strong>of</strong><br />
X|V ∼Po(λ = V ).<br />
Find E(X) and Var(X).<br />
Solution. Use<br />
E(X) = E{E(X|V )}<br />
Var(X) = E V {Var(X|V )} + Var V {E(X|V )}<br />
E(X|V ) = V<br />
Var(X|V ) = V<br />
so E(X) = E{E(X|V )} = E(V ) = α λ .<br />
Var(X) = E V {Var(X|V )} + Var V {E(X|V )}<br />
= E V (V ) + Var V (V )<br />
= α λ + α λ 2<br />
= α λ<br />
(<br />
1 + 1 ) ( ) 1 + λ<br />
= α .<br />
λ λ 2<br />
52