Title
Final summary
Week 12 lecture
Title
Chapter 3. Special distribution
Important techniques and facts
Moment generating technique
Binomial and multinomial: joint, conditional, marginal
distribution, variance and covariance structure, MGF
Poisson: co
(1) Use the method of integration by part and induction to prove the following equality.
w
k1
(w)x ew
z k1 ez
dz =
.
(k 1)!
x!
x=0
(2) Given Poisson random variables X1 , . . . , Xn , each Xi P (mi ), Y = n Xi , prove that
i=1
Y P ( n mi ).
i=1
3: Three
i
1-\ \IV
S o(U?h~oY\.
P e_ ~ (@n -f)bl R!'~'f) ~ o
mv
d
()~ r T
/\
e~ -eo
p
-:-J.
-I
cfw_'
p
o , ~ t> e;.,p ~'ab ~' (\cfw_)~-e.J -7 o
Acc 01ot: ~ :;(,-, ~ vJC'v 'otcJ Cm d.t: 0'11 R5"
d'>lo81cx:; &J ~ M (.:x:,\ E6 , (M W)<Oo ~ tW(
v
2
-'-?'
-oe>
E>o- c <
(
)
Indep. ! P Ai1 Ai2 . Ai j =
j
n=1
( P ( A )
Multivariate Distribution (Two Random Variables)
Discrete Dist. Def.: Space is finite or countable
pmf: ! p X1 ,X2 ( x1 , x2 ) = P X1 = x1 , X 2 = x2
in
[
! P ( X , X ) B = p
Marginal pmf: ! p ( x ) =
! E (