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 MultiStage 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 MultiStage
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 nStage independent Bernoulli Trial then the probability that will occur exactly times is
referred to as a Binomial Distribution.
Moreover the Expected Value of is
The formula:

Properties:
Since



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):
Setting
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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