Home >>STATISTICS, Section 2, binomial distribution 1
Definition |
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.
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.
Notation B(n, p)
The full notation describing a Binomial distribution is:
where,
X is a random variable (0, 1, 2, 3,...)
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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,
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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