Probability

The definition of "rare" is subjective and depends heavily on the context in which it is used.

Here's a breakdown of how "rare" can be interpreted:

• General Usage: In everyday language, "rare" typically means something not commonly found or seen; something unusual or exceptional.
• Statistical Context: In statistics, a rare event is one with a low probability of occurring. The threshold for what constitutes a "low probability" is often arbitrary and depends on the specific application. For example, in hypothesis testing, a p-value of less than 0.05 (5%) is often considered statistically significant and might be described as rare.
• Specific Fields:
• Medicine: A rare disease is generally defined as a condition that affects a small percentage of the population. In the United States, this is typically defined as affecting fewer than 200,000 people. Source
• Ecology: A rare species is one that has a small population size, a restricted geographic range, or both. Source
• Collectibles: In the world of collectibles (stamps, coins, cards, etc.), rarity is a key factor in determining value. A rare item is one with few known examples, often due to limited production or accidental destruction.

In summary, there's no single, universally accepted definition of "rare." Its meaning is relative and must be understood within its specific context.

A Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known constant mean rate and independently of the time since the last event. Its characteristics include:

1. Discrete Data: It deals with discrete data, meaning the data can only take certain values (typically non-negative integers).
2. Independence: Events are independent; the occurrence of one event does not affect the probability of another event occurring.
3. Constant Mean Rate: The average rate at which events occur is constant over the given interval. This rate is denoted by λ (lambda).
4. Randomness: Events occur randomly and uniformly throughout the interval.
5. Non-Negative Integer Values: The distribution gives the probability of observing 0, 1, 2, 3, ... events.
6. Well-Defined Probability Mass Function: The probability of observing exactly k events is given by the formula:

   P(X = k) = (λ^k * e^(-λ)) / k!

   where:

   • λ is the average rate of events.
   • e is Euler's number (approximately 2.71828).
   • k is the number of events.
   • k! is the factorial of k.
7. Mean and Variance: The mean (average) and the variance of a Poisson distribution are both equal to λ.
8. Applications: It's often used to model rare events such as:

   • The number of phone calls received by a call center per hour.
   • The number of cars passing a point on a highway per minute.
   • The number of defects in a manufactured product.

No, a square matrix A is not invertible if its determinant |A| is equal to 0. A matrix is invertible (also known as non-singular or non-degenerate) if and only if its determinant is non-zero. If the determinant is zero, the matrix is singular and does not have an inverse.

Invertibility requires that the matrix represents a transformation that can be "undone." When the determinant is zero, it means the matrix collapses space (or at least reduces its dimension), making it impossible to reverse the transformation uniquely.

You can explore more about invertible matrices and their properties on websites such as:

• MathWorld: Invertible Matrix
• Wikipedia: Invertible Matrix

The term "wild man" has a few different interpretations and historical contexts:

• Mythological/Folklore Creature: In European folklore, the "wild man" (also known as a "woodwose" or "wudewasa") is a mythical figure resembling a hairy, primitive human or hominid, often associated with the forest. They were typically depicted as living outside of civilization and embodying untamed nature.
• Historical Encounters: Throughout history, there have been accounts of individuals living in the wilderness, isolated from society. These accounts sometimes led to the "wild man" label, although these were cases of feral children or hermits who had chosen to live apart from others.
• Figurative Use: The term can also be used figuratively to describe someone with uncivilized or savage behavior.