For now we are just considering of a simpler experiment, transmitting a 0 or transmitting a 1 with the possibility of error.
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A Bernoulli Trial is an experiment with exactly two possible
outcomes, say
and
.
If the probability of
occuring
is
,
then the probability of
occuring
is
.
A Multi-Stage independent Bernoulli Trial
consists of performing the same Bernoulli Trial more than once,
assuming that performing each stage does not affect the outcomes of those that
follows.
Formally, an
-Stage
independent Bernoulli Trial is a Multi-Stage
independent Bernoulli Trial in which
is the number of times the experiment was performed.
The Sample Space is the consists of all sequences
where
or
.
We will also want to look at the associated Random Variable
Count
of occurences of
Using the notation above, suppose one performs an n-Stage independent
Bernoulli Trial then the probability that
will
occur exactly
times
is
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referred to as a Binomial Distribution.
Moreover the Expected Value of
is
The formula:
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Properties:
Since
|
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For any numbers
and
:
In
particular,
And
The Calculation:
There are two cases to consider:
is not an integer:
for
so
for
so
Hence the maximum is an integer near
is an integer:
,
the maximums.
Theorem :
Proof (the hard way):
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Setting
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Proof (the easy way):
Exercise: Show if you do the Trial once the expected number
of occurences is
Review
the concept of independent events , in particular that the
expected outcome of independent events is the sum of the expected
outcome of the individual events.
Theorem:
Suppose a given a Bernoulli Trial with possible outcomes
and
and
can
be the experiment in an
-Stage
independent Bernoulli Trial for any
.
Let
be the number of times
that the outcome is
in a given
-Stage
independent Bernoulli Trial. Then
for any
See Bernoulli Trials from the Center for Imaging Science, RIT
Suppose one transmits a Bit and the probability of transmission error is
.
As a strategy, a way of improving the chance of correctly transmitting the Bit
is to transmit it 3 times and choose the Bit that comes most
often. Now the probability of error is
For
example if
then
the "2 out of 3" probability is
.
If this is not good enough transmit it 5 times. This is not a
very efficient strategy.
Exercise: Why does this strategy work?
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