STAT 333
Fall 2010
October 18, 2010
[Thursday, Sep 30, 2010]
Objective: Conditional expectation and doubleaverage theorem
Example 1
Consider i.i.d. Bernoulli trials with
p
r
(
S
) =
p,
0
< p <
1. Define that
T
s
= the
waiting time for the first
S
, and
T
ss
= the waiting time for the first ”
SS
” in a row. Then
T
s
∼
Geo
(
p
). Find E(
T
s
) and E(
T
SS
).
Example 2
A miner is trapped in a mine. There are three doors in front of the miner. The
first door leads to a tunnel that takes him to safety after two hours of travel. The second
door leads to a tunnel that returns him to the mine after three hours of travel. The third
door leads to a tunnel that returns him to his mine after five hours.
Assuming that the
miner is at all times equally likely to choose any one of the doors. Let
X
= the number of
hours to reach the safe place.
1. Find E(X) and Var(X).
2. If the first door leads the miner to the safe place after
n
1
hours, and the other two doors
lead the miner to the safe place after
n
2
and
n
3
hours, respectively. Find E(X) and Var(X).
Example 3
Let
N
be the number of total claims to an insurance company in 2011. Sup
pose that
N
follows a Poisson distribution with rate
λ
, such that
N
∼
Poi
(
λ
).
Denote
X
i
, i
= 1
,
2
, ...
the amount payed to the
i
th claim. We assume that
N
and
X
i
, i
= 1
,
2
, ...
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 Fall '10
 MenZhongxian
 Bernoulli, Probability, Probability theory, insurance company

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