Bernoulli Trials

A Binary Symmetric Channel - where we are going:

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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Bernoulli Trials:

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

referred to as a Binomial Distribution.

Moreover the Expected Value of is

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Binomial Coefficients:

The formula:

Properties:

1. 

2. 

   Since

   (

   (

   

3. For any numbers and :

In particular,

And

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The Maximum Term of a Binomial Distribution:

The Calculation:

There are two cases to consider:

1. is not an integer:

   for so

   for so

   Hence the maximum is an integer near

2. is an integer:

   , the maximums.

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The Expected Number of Occurences of :

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.

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The Law of Large Numbers:

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

or

See Bernoulli Trials from the Center for Imaging Science, RIT

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Majority Rules Error Detection:

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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