Statistics: Understanding Statistical Distributions: Bernoulli, Binomial, Negative Binomial, Geometric, Hypergeometric, Poisson

BERNOULLI TRIALS

Bernoulli trials refer to a sequence of independent experiments/events that can result in only two possible outcomes, often referred to as “success” and “failure.”

1. Binary Outcomes: In Bernoulli trials, the outcomes are binary, meaning they can be classified into two distinct categories, such as “success” and “failure,” “yes” and “no,” or “1” and “0.” These outcomes can be represented numerically or categorically, depending on the data and the analysis being conducted.

2. Independent trials: Each trial in a sequence of Bernoulli trials must be independent of the others. This implies that the outcome of one trial does not influence or affect the outcome of any other trial in the sequence. This assumption of independence is crucial for many statistical models and analyses.

3. Fixed Probability of Success: In Bernoulli trials, the probability of success (often denoted as “p”) remains constant throughout the sequence of trials. The probability of failure is then given by (1 — p). It is important to note that the probability of success remains the same for each trial and is not influenced by previous outcomes.

Modeling Binary Events: Bernoulli trials are often used to model binary events or phenomena, where the occurrence or non-occurrence of an event is of interest. For example, in data science, Bernoulli trials can be employed to model the success or failure of a marketing campaign, the presence or absence of a specific feature in a dataset, or the occurrence or non-occurrence of an event in a time series.

Bernoulli trials serve as the foundation for various statistical models and techniques in data science. They are essential components of many probability distributions and statistical tests. For instance, the Bernoulli distribution is a discrete probability distribution that describes the probability of a single success or failure in a single Bernoulli trial. It is often used in logistic regression, which is a widely employed classification algorithm in data science.

Extension to Multiple Trials: While a single Bernoulli trial consists of one event, the concept can be extended to multiple trials, forming the basis for other important probability distributions such as the binomial distribution, which…