STATISTICS - Section 2

 

Binomial Distribution 1

 

 

Definition

Notation

Limits

 

 

 

Definition

 

The Binomial Distribution describes the behavior of a random variable(count variable)  X under the following conditions:

 

 

1. The number of trials(n) is fixed.


2. Trials have only 2 possible outcomes (success/failure).


3. Each trial is independent of the other.


4. Probability of success(p) is constant throughout.

 

 

X(the random variable) is a measure of the number of successes in n trials.

 

 

 

 

Example

 

A simple example is choosing 1 ball from a bag of 10 identical balls, each numbered (1-10).

 

Once noted, the ball is returned to the bag.


A single ball is chosen on 3 separate occasions.

 

Success is in obtaining a '5' ball.

 

 

So the random variable X has values 0, 1, 2, 3 .

 

In other words, from our 3 tries we could have obtained:

 

0 fives,    1 five,   2 fives,   3 fives

 

On the first try, the probability of obtaining a 5 is 1/10 .


The probability of not getting a 5 is 9/10 .

 

 

Every time we dip into the bag of 10 balls, the probability of obtaining a 5 is 1/10.

 

The probability is constant.

 

Getting a '5' or not getting a '5' means that there are only 2 outcomes.

 

Every try is independent.

 

Previous tries do not affect the result, since previously chosen balls are returned to the bag.

 

Every try is taken from the bag of 10 balls.

 

 

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Notation B(n, p)

 

The full notation describing a Binomial distribution is:

 

 

 

where,

 

X is a random variable (0, 1, 2, 3,...)


~ B   'is distributed Binomially'


n is the number of trials


p is the probability of a single trial 'success'

 

 

 

 

 

Example (continued from above)

 

Say that there are only 3 tries of attempting to take a 5-ball from a bag of 10 balls.

 

So n = 3 and p = 1/10.

 

The possible number of 5's taken in the 3 trials is summarized by the values of the random variable X .

 

X = 0, 1, 2, 3

 

Using the Binomial notation,

 

 

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Limits

 

The population size(n) of a Binomial Distribution must be much larger than the sample size(r).

 

The distribution only applies to trials from a simple random sample.

 

Under this condition n must be at least x10 times > r .

 

Outside this limit, the equation is not valid.

 

 

 

 

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