Chapter 4. Probability Distributions: Discrete Probability Distributions
The Bernoulli Probability Distribution
Bernoulli Trial
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#:
\[\begin{array}{lcrcl}
\mathbb{P}(\text{success})\,=\,p&\phantom{Batman}&\mathbb{P}(\text{failure})&=&q\\\\
&&&=&1-p
\end{array}\]
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 Bernoulli random variable.
We say that #X# is Bernoulli distributed with parameter #p# and write this as: \[X \sim Bernoulli(p)\]
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 = 0.50#
The expected value of a Bernoulli random variable #X\sim Bernoulli(p)# is calculated as follows:
\[\begin{array}{rcl}
\mu&=&p\\\\
&=& 0.50
\end{array}\]
The expected value of a Bernoulli random variable #X\sim Bernoulli(p)# is calculated as follows:
\[\begin{array}{rcl}
\mu&=&p\\\\
&=& 0.50
\end{array}\]
The Bernoulli distribution serves as the foundation for two other discrete probability distributions:
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