Random variableA random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. The term 'random variable' can be misleading as it is not actually random nor a variable, but rather it is a function from possible outcomes (e.g., the possible upper sides of a flipped coin such as heads and tails ) in a sample space (e.g., the set ) to a measurable space (e.g., in which 1 corresponding to and −1 corresponding to ), often to the real numbers.
HatA hat is a head covering which is worn for various reasons, including protection against weather conditions, ceremonial reasons such as university graduation, religious reasons, safety, or as a fashion accessory. Hats which incorporate mechanical features, such as visors, spikes, flaps, braces or beer holders shade into the broader category of headgear. In the past, hats were an indicator of social status. In the military, hats may denote nationality, branch of service, rank or regiment.
Top hatA top hat (also called a high hat, a cylinder hat, or, informally, a topper) is a tall, flat-crowned hat traditionally associated with formal wear in Western dress codes, meaning white tie, morning dress, or frock coat. Traditionally made of black silk or sometimes grey, the top hat emerged in Western fashion by the end of the 18th century. Although it declined by the time of the counterculture of the 1960s, it remains a formal fashion accessory. A collapsible variant of a top hat, developed in the 19th century, is known as an opera hat.
Exchangeable random variablesIn statistics, an exchangeable sequence of random variables (also sometimes interchangeable) is a sequence X1, X2, X3, ... (which may be finitely or infinitely long) whose joint probability distribution does not change when the positions in the sequence in which finitely many of them appear are altered. Thus, for example the sequences both have the same joint probability distribution. It is closely related to the use of independent and identically distributed random variables in statistical models.
Σ-finite measureIn mathematics, a positive (or signed) measure μ defined on a σ-algebra Σ of subsets of a set X is called a finite measure if μ(X) is a finite real number (rather than ∞), and a set A in Σ is of finite measure if μ(A) < ∞. The measure μ is called σ-finite if X is a countable union of measurable sets each with finite measure. A set in a measure space is said to have σ-finite measure if it is a countable union of measurable sets with finite measure. A measure being σ-finite is a weaker condition than being finite, i.