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In statistics, a circumflex (ˆ), called a "hat", is used to denote an estimator or an estimated value. [1] For example, in the context of errors and residuals, the "hat" over the letter ^ indicates an observable estimate (the residuals) of an unobservable quantity called (the statistical errors).
In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in ^ (pronounced "v-hat"). The term normalized vector is sometimes used as a synonym for unit vector.
Greek letters (e.g. θ, β) are commonly used to denote unknown parameters (population parameters). [3]A tilde (~) denotes "has the probability distribution of". Placing a hat, or caret (also known as a circumflex), over a true parameter denotes an estimator of it, e.g., ^ is an estimator for .
Functional notation: if the first is the name (symbol) of a function, denotes the value of the function applied to the expression between the parentheses; for example, (), (+). In the case of a multivariate function , the parentheses contain several expressions separated by commas, such as f ( x , y ) {\displaystyle f(x,y)} .
In English, the circumflex, like other diacritics, is sometimes retained on loanwords that used it in the original language (for example entrepôt, crème brûlée). In mathematics and statistics, the circumflex diacritic is sometimes used to denote a function and is called a hat operator.
Greek letters are used in mathematics, science, engineering, and other areas where mathematical notation is used as symbols for constants, special functions, and also conventionally for variables representing certain quantities. In these contexts, the capital letters and the small letters represent distinct and unrelated entities.
In mathematics, x̂ often refers to the unit vector in the +X direction. In statistics, it can also represent the predicted value for x, consistent with the hat notation . [ 3 ]
In this case the problem reduces to n − 2 people and n − 2 hats, because P 1 received h i ' s hat and P i received h 1 's hat, effectively putting both out of further consideration. For each of the n − 1 hats that P 1 may receive, the number of ways that P 2, ..., P n may all receive hats is the sum of the counts for the two cases.