# Statistics/Distributions/Bernoulli

### Bernoulli Distribution: The coin toss

Parameters ${\displaystyle 0 ${\displaystyle k=\{0,1\}\,}$ ${\displaystyle {\begin{cases}q=(1-p)&{\text{for }}k=0\\p&{\text{for }}k=1\end{cases}}}$ ${\displaystyle {\begin{cases}0&{\text{for }}k<0\\q&{\text{for }}0\leq k<1\\1&{\text{for }}k\geq 1\end{cases}}}$ ${\displaystyle p\,}$ ${\displaystyle {\begin{cases}0&{\text{if }}q>p\\0.5&{\text{if }}q=p\\1&{\text{if }}q ${\displaystyle {\begin{cases}0&{\text{if }}q>p\\0,1&{\text{if }}q=p\\1&{\text{if }}q ${\displaystyle p(1-p)\,}$ ${\displaystyle {\frac {q-p}{\sqrt {pq}}}}$ ${\displaystyle {\frac {1-6pq}{pq}}}$ ${\displaystyle -q\ln(q)-p\ln(p)\,}$ ${\displaystyle q+pe^{t}\,}$ ${\displaystyle q+pe^{it}\,}$ ${\displaystyle q+pz\,}$ ${\displaystyle {\frac {1}{p(1-p)}}}$

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:

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

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:

${\displaystyle \operatorname {E} [X]=\sum _{i}f(x_{i})\cdot x_{i}}$
${\displaystyle \operatorname {E} [X]=p\cdot 1+(1-p)\cdot 0}$
${\displaystyle \operatorname {E} [X]=p\,}$

#### Variance

${\displaystyle \operatorname {Var} (X)=\operatorname {E} [(X-\operatorname {E} [X])^{2}]=\sum _{i}f(x_{i})\cdot (x_{i}-\operatorname {E} [X])^{2}}$
${\displaystyle \operatorname {Var} (X)=p\cdot (1-p)^{2}+(1-p)\cdot (0-p)^{2}}$
${\displaystyle \operatorname {Var} (X)=[p(1-p)+p^{2}](1-p)\,}$
${\displaystyle \operatorname {Var} (X)=p(1-p)\,}$