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Statistics/Distributions/Bernoulli

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Bernoulli Distribution: The coin tossEdit

Bernoulli
Parameters  
Support  
PMF  
CDF  
Mean  
Median  
Mode  
Variance  
Skewness  
Ex. kurtosis  
Entropy  
MGF  
CF  
PGF  
Fisher information  

There is no more basic random event than the flipping of a coin. Heads or tails. It's as simple as you can get! The "Bernoulli Trial" refers to a single event which can have one of two possible outcomes with a fixed probability of each occurring. You can describe these events as "yes or no" questions. For example:

  • Will the coin land heads?
  • Will the newborn child be a girl?
  • Are a random person's eyes green?
  • Will a mosquito die after the area was sprayed with insecticide?
  • Will a potential customer decide to buy my product?
  • Will a citizen vote for a specific candidate?
  • Is an employee going to vote pro-union?
  • Will this person be abducted by aliens in their lifetime?

The Bernoulli Distribution has one controlling parameter: the probability of success. A "fair coin" or an experiment where success and failure are equally likely will have a probability of 0.5 (50%). Typically the variable p is used to represent this parameter.

If a random variable X is distributed with a Bernoulli Distribution with a parameter p we write its probability mass function as:

 

Where the event X=1 represents the "yes."

This distribution may seem trivial, but it is still a very important building block in probability. The Binomial distribution extends the Bernoulli distribution to encompass multiple "yes" or "no" cases with a fixed probability. Take a close look at the examples cited above. Some similar questions will be presented in the next section which might give an understanding of how these distributions are related.

MeanEdit

The mean (E[X]) can be derived:

 
 
 

VarianceEdit

 
 
 
 

External linksEdit