Modelling a random variable with the Bernoulli Distribution

In many real-world situations, we encounter scenarios where the outcome is uncertain. However, we can use mathematical models to describe this uncertainty and improve our understanding of these uncertain outcomes. To produce these models, we will need to use the following mathematical tools:

A random variable is a mathematical way of representing the outcome of this uncertain phenomenon numerically. It's a variable whose value is a numerical result of a process that has some element of chance or uncertainty.
Separately, a probability distribution describes how likely different values of a random variable are. It's a mathematical function that tells you what the probability is for each possible outcome of a random variable. Probability distributions may also have parameters, which change the probabilities assigned to each outcome of the random variable. We can also use metrics like the mean and variance to produce a numerical summary of a distribution.
Finally, a statistic is a numerical value which is calculated from a sample of data. It's a way to summarize or describe a particular characteristic of that sample. Unlike a parameter like $p$ , which describes a distribution, a statistic describes only the sample. Because samples are measured from uncertain observations, they too are random variables and will vary from sample to sample. We often use statistics to estimate or test hypotheses about distributions and their parameters.

When we are modelling a random process, we must first define the relevant random variables and then decide on an appropriate probability distribution and parameters to describe the probabilities associated with the outcomes for these random variables. It is important to note that we cannot see or measure a distribution parameter, we can only estimate it from the data we observe. We can then measure statistics on smaples of data from the random process to determine if this distribution and its parameters are indeed appropriate.

The simplest random process is one with two discrete outcomes - success or failure. To model this, we need a random variable $X$ which takes a numerical value depending on the outcome. We will assign 'success' a value of $1$ , and 'failure' a value of $0$ , so we could think of this random variable $X$ as describing 'the number of successes observed'. This random variable is called a Bernoulli random variable.

The Bernoulli distribution is a probability distribution that models a Bernoulli random variable $X$ . The distribution has a parameter $p$ which defines the probability of success. Thus, the definition of this distribution is

P(X = x) = \begin{cases} p, & x = 1, \quad \text{(success)} \\ 1 - p, & x = 0, \quad \text{(failure)} \end{cases}

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