1
Whither the Binomial…
•
Recall example of sending bit string over network
n
= 4 bits sent over network where each bit had
independent probability of corruption
p
= 0.1
X = number of bit corrupted.
X ~ Bin(4, 0.1)
In real networks, send large bit strings (length
n
10
4
)
Probability of bit corruption is very small
p
10
-6
X ~ Bin(10
4
, 10
-6
) is unwieldy to compute
•
Extreme
n
and
p
values arise in many cases
# bit errors in file written to disk (# of typos in a book)
# of elements in particular bucket of large hash table
# of servers crashes in a day in giant data center
# Facebook login requests that go to particular server
Binomial in the Limit
•
Recall the Binomial distribution
•
Let
l
=
np
(equivalently:
p
=
l
/
n
)
•
When
n
is large,
p
is small, and
l
is “moderate”:
•
Yielding:
i
n
i
p
p
i
n
i
n
i
X
P
)
1
(
)!
(
!
!
)
(
i
n
i
i
i
n
i
n
n
i
n
n
n
i
n
i
n
i
X
P
i
n
n
n
)
/
1
(
)
/
1
(
!
1
)!
(
!
!
)
(
)
1
)...(
1
(
l
l
l
l
l
l
l
e
n
n
)
/
1
(
1
)
1
)...(
1
(
i
n
i
n
n
n
1
)
/
1
(
i
n
l
l
l
l
l
e
i
e
i
i
X
P
i
i
!
1
!
1
)
(
Poisson Random Variable
•
X is a
Poisson
Random Variable:
X ~ Poi(
l
)
X takes on values 0, 1, 2…
and, for a given parameter
l
> 0,
has distribution (PMF):
•
Note Taylor series:
•
So:
!
)
(
i
e
i
X
P
i
l
l
0
2
1
0
!
...
!
2
!
1
!
0
i
i
i
e
l
l
l
l
l
1
!
!
)
(
0
0
0
l
l
l
l
l
l
e
e
i
e
i
e
i
X
P
i
i
i
i
i
Sending Data on Network Redux
•
Recall example of sending bit string over network
Send bit string of length
n
= 10
4
Probability of (independent) bit corruption
p
= 10
-6
X ~ Poi(
l
= 10
4
* 10
-6
= 0.01)
What is probability that message arrives uncorrupted?
Using Y ~ Bin(10
4
, 10
-6
):
990049834
.
0
!
0
)
01
.
0
(
!
)
0
(
0
01
.
0
e
i
e
X
P
i
l
l
990049829
.
0
)
0
(
Y
P
Caveat emptor: Binomial computed with built-in function in R software
package, so some approximation may have occurred.
Approximation
are closer to you than they may appear in some software packages.

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