If A and B are independent events, the probability of both occurring is P(A ∩ B) which is equal to P(A) * P(B). This is the fundamental characteristic of independent events in probability.

The power of a test is the ability of the test to detect an effect if it truly exists. It's the probability that the test correctly rejects a false null hypothesis. High power is desirable because it means the test is less likely to make a Type II error (false negative). When designing a study, it's important to choose a sample size and significance level that will provide enough power to detect an effect if one exists.

Standard deviation is a measure of how spread out the numbers in a dataset are around the mean. It measures the average distance between each data point and the mean. The higher the standard deviation, the more spread out the data is.

The centroid is the geometric center of a cluster. In other words, it's the mean value of all the points in a specific cluster.

Monotonic transformations do not affect the Spearman’s rank correlation coefficient. This is because Spearman's correlation is based on the rank order of data, and monotonic transformations preserve this order.

The alternative hypothesis proposes that there is a difference or an effect in the population and the sample observations are influenced by some non-random cause.

To create a user-defined exception, one must usually extend the Exception class. By extending this base class, developers can define their own custom exceptions with specific behaviors, making the code more modular and maintainable.

JSON (JavaScript Object Notation) is commonly used for transmitting data in social media integrations and APIs due to its lightweight and human-readable format. It is easy to parse and generate, making it a popular choice for data exchange.

The 'application' directory in CodeIgniter is primarily used for configuration and customization. It houses files related to controllers, models, views, and other application-specific components. This directory is crucial for organizing and managing the core functionality of a CodeIgniter application.

The load function in CodeIgniter is used to load libraries, helpers, or custom config files within a controller. This enables the use of additional resources, enhancing the functionality of the controller.