ISyE 2027 Probability with Applications
Polly B. He
Fall10, Week 11
Reading: Section 9.4
Independent Random Variables
Recall the independence of events. We say two events
A
and
B
are independent if
P
(
A
∩
B
) =
P
(
A
)
P
(
B
).
What about the independence between random variables?
Quick exercise: do you think the following random variables are independent?
1. We toss a fair coin once and then roll a die. Let
X
be 1 if the coin comes up as a head
and 0 otherwise. Let
Y
be the outcome of the die.
2. The price movements of corn and soy beans on the Chicago Mercantile Exchange.
Deﬁnition: Two random variables
X
and
Y
, with joint distribution function
F
, are
indepen
dent
if
P
(
X
≤
a,Y
≤
b
) =
P
(
X
≤
a
)
P
(
Y
≤
b
)
,
i.e.,
F
(
a,b
) =
F
X
(
a
)
F
Y
(
b
)
,
for all
∞
< a,b <
∞
Further, checking the last equation is equivalent to checking one of the following statements:
1. For all sets
A
and
B
,
P
(
X
∈
A,Y
∈
B
) =
P
(
X
∈
A
)
P
(
Y
∈
B
)
2. If
X
and
Y
are both discrete random variables,
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 Fall '08
 Zahrn
 Probability theory, probability density function, Chicago Mercantile Exchange, Polly B. He Fall10

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