1
Independent Discrete Variables
•
Two discrete random variables X and Y are
called
independent
if:
•
Intuitively: knowing the value of X tells us nothing
about the distribution of Y (and vice versa)
If two variables are
not
independent, they are called
dependent
•
Similar conceptually to independent
events
, but
we are dealing with multiple
variables
Keep your events and variables distinct (and clear)!
y
x
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p
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Coin Flips
•
Flip coin with probability
p
of “heads”
Flip coin a total of
n
+
m
times
Let X = number of heads in first
n
flips
Let Y = number of heads in next
m
flips
X and Y are independent
Let Z = number of total heads in
n
+
m
flips
Are X and Z independent?
o
What if you are told Z = 0?
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p
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X
P
)
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P
Web Server Requests
•
Let N = # of requests to web server/day
Suppose N ~ Poi(
l
)
Each request comes from a human (probability =
p
) or
from a “bot” (probability = (1 –
p
)), independently
X = # requests from humans/day
(X  N) ~ Bin(N,
p
)
Y = # requests from bots/day
(Y  N) ~ Bin(N, 1 
p
)
Note:
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Web Server Requests (cont.)
•
Let N = # of requests to web server/day
Suppose N ~ Poi(
l
)
Each request comes from a human (probability =
p
) or
from a “bot” (probability = (1 –
p
)), independently
X = # requests from humans/day
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 Spring '09
 Probability theory, probability density function, Web server

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