# Statistics/Distributions/Bernoulli

## Contents

### Bernoulli Distribution: The coin toss

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

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

$\operatorname{Var}(X) = \operatorname{E}[(X-\operatorname{E}[X])^2] = \sum_i f(x_i) \cdot (x_i - \operatorname{E}[X])^2$
$\operatorname{Var}(X)= p \cdot (1-p)^2 + (1-p) \cdot (0-p)^2$
$\operatorname{Var}(X)= [p(1-p) + p^2](1-p) \,$
$\operatorname{Var}(X)= p(1-p) \,$