Statistics/Distributions/Bernoulli

Bernoulli Distribution: The coin toss

Bernoulli

Parameters	$0<p<1,p\in \mathbb {R}$
Support	$k=\{0,1\}\,$
PMF	${\begin{cases}q=(1-p)&{\text{for }}k=0\\p&{\text{for }}k=1\end{cases}}$
CDF	${\begin{cases}0&{\text{for }}k<0\\q&{\text{for }}0\leq k<1\\1&{\text{for }}k\geq 1\end{cases}}$
Mean	$p\,$
Median	${\begin{cases}0&{\text{if }}q>p\\0.5&{\text{if }}q=p\\1&{\text{if }}q<p\end{cases}}$
Mode	${\begin{cases}0&{\text{if }}q>p\\0,1&{\text{if }}q=p\\1&{\text{if }}q<p\end{cases}}$
Variance	$p(1-p)\,$
Skewness	${\frac {q-p}{\sqrt {pq}}}$
Ex. kurtosis	${\frac {1-6pq}{pq}}$
Entropy	$-q\ln(q)-p\ln(p)\,$
MGF	$q+pe^{t}\,$
CF	$q+pe^{it}\,$
PGF	$q+pz\,$
Fisher information	${\frac {1}{p(1-p)}}$

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:

$f(x)={\begin{cases}p,&{\mbox{if }}x=1\\1-p,&{\mbox{if }}x=0\end{cases}}\quad 0\leq p\leq 1$

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

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

$\operatorname {E} [X]=\sum _{i}f(x_{i})\cdot x_{i}$

$\operatorname {E} [X]=p\cdot 1+(1-p)\cdot 0$

$\operatorname {E} [X]=p\,$

Variance

$\operatorname {Var} (X)=\operatorname {E} [(X-\operatorname {E} [X])^{2}]=\sum _{i}f(x_{i})\cdot (x_{i}-\operatorname {E} [X])^{2}$

$\operatorname {Var} (X)=p\cdot (1-p)^{2}+(1-p)\cdot (0-p)^{2}$

$\operatorname {Var} (X)=[p(1-p)+p^{2}](1-p)\,$

$\operatorname {Var} (X)=p(1-p)\,$

External links

• Interactive Bernoulli Distribution Web Applet (Java)