Probability Distribution

A probability distribution is a way of describing how likely different outcomes are in a particular
situation. Imagine you are flipping a fair coin. The probability distribution would tell you the chances of getting heads or tails. It provides a way to understand the likelihood of different events or results.

Think of it like a menu at a restaurant. Each item on the menu represents a possible outcome, and the prices next to them indicate their probabilities. Some dishes may be more likely to be ordered, so they
have higher probabilities, while others may be less likely, so they have lower probabilities.

Probability distributions can have different shapes and patterns. For example, the distribution for rolling
a fair six-sided die would be uniform, meaning each outcome has an equal chance of occurring. On the
other hand, the distribution for heights of adult males might follow a bell-shaped curve called the
normal distribution, where most men fall close to the average height.

These distributions help us understand and analyze data, make predictions, and make informed
decisions. By studying the probability distribution of a situation, we can gain insights into the likelihood
of specific events and estimate the chances of different outcomes occurring.

There are several types of probability distributions, each suited for different scenarios and applications.
Some of the commonly encountered distributions are:

Uniform Distribution

This distribution assigns equal probabilities to all outcomes within a specified range. For example, rolling a fair die or selecting a random number from 1 to 10.

Binomial Distribution

It models the number of successes in a fixed number of independent Bernoulli trials. It is useful when dealing with binary outcomes (e.g., success/failure, yes/no) and requires parameters such as the probability of success and the number of trials.

Normal Distribution

Also known as the Gaussian distribution, it is characterized by its bell-shaped curve. Many natural phenomena, such as heights or test scores, tend to follow this distribution due to the central limit theorem. It is defined by its mean and standard deviation.

Poisson Distribution

It models the number of events that occur within a fixed interval of time or space. It is commonly used to describe rare events or phenomena with a low probability of occurrence.

Exponential Distribution

This distribution is often used to model the time between events in a Poisson
process. It describes the probability of waiting a certain amount of time before the next event occurs.

Gamma Distribution

It is a versatile distribution that extends the exponential distribution and can model various continuous positive outcomes.

Beta Distribution

It is a continuous probability distribution defined on the interval [0, 1]. It is commonly used as a prior distribution for probabilities in Bayesian statistics.