### Chapter 4. Probability Distributions: Discrete Probability Distributions

### The Bernoulli Probability Distribution

Bernoulli Trial

A

\[\Omega=\{\text{success}, \text{ failure}\}\] Typically, a success is encoded as a '#1#' and a failure as a '#0#'.

In a Bernoulli trial, the probability of observing a success is denoted by #p# and the probability of failure is denoted by #q#:

\[\begin{array}{lcrcl}

\mathbb{P}(\text{success})\,=\,p&\phantom{Batman}&\mathbb{P}(\text{failure})&=&q\\\\

&&&=&1-p

\end{array}\]

A

**Bernoulli trial**is a random experiment with two possible outcomes:*success*or*failure*.\[\Omega=\{\text{success}, \text{ failure}\}\] Typically, a success is encoded as a '#1#' and a failure as a '#0#'.

In a Bernoulli trial, the probability of observing a success is denoted by #p# and the probability of failure is denoted by #q#:

\[\begin{array}{lcrcl}

\mathbb{P}(\text{success})\,=\,p&\phantom{Batman}&\mathbb{P}(\text{failure})&=&q\\\\

&&&=&1-p

\end{array}\]

Bernoulli Random Variable

Let #X# be a random variable that takes on a value of #1# with probability #p# and a value of #0# with probability #1-p#, then #X# is a

We say that #X# is

Let #X# be a random variable that takes on a value of #1# with probability #p# and a value of #0# with probability #1-p#, then #X# is a

**Bernoulli****random variable**.We say that #X# is

**Bernoulli distributed**with parameter #p# and write this as: \[X \sim Bernoulli(p)\]Suppose we roll a regular six-sided die and define a success as rolling an even number.

Let #X# be a random variable that takes on a value of #1# when we roll an even number and a value #0# when we roll an uneven number.

Then #X# is *Bernoulli *distributed with #p=\mathbb{P}(\text{Roll an even number}) = 0.5#: \[X\sim Bernoulli(0.5)\]

#\text{}#

Mean and Standard Deviation of a Bernoulli Random Variable

Let #X# be a *Bernoulli* distributed random variable with parameter #p#.

Then the **expected value** of #X# calculated with the following formula: \[\mu = p\]

The **variance **of #X# is calculated with the following formula:\[\sigma^2 = p \cdot (1-p)\]

And the **standard deviation **of #X# is calculated with the following formula:\[\sigma = \sqrt{p \cdot (1-p)}\]

#\mu = 0.25#

The

\[\begin{array}{rcl}

\mu&=&p\\\\

&=& 0.25

\end{array}\]

The

*expected value*of a Bernoulli random variable #X\sim Bernoulli(p)# is calculated as follows:\[\begin{array}{rcl}

\mu&=&p\\\\

&=& 0.25

\end{array}\]

The Bernoulli distribution serves as the foundation for two other discrete probability distributions:

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