Statistics/Distributions/Bernoulli

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Bernoulli Distribution: The coin toss[edit]

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.

Mean[edit]

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

Variance[edit]

External links[edit]