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When "E" is used to denote "expected value", authors use a variety of stylizations: the expectation operator can be stylized as E (upright), E (italic), or (in blackboard bold), while a variety of bracket notations (such as E(X), E[X], and EX) are all used. Another popular notation is μ X.
The global maximum of x √ x occurs at x = e. Steiner's problem asks to find the global maximum for the function =. This maximum occurs precisely at x = e. (One can check that the derivative of ln f(x) is zero only for this value of x.) Similarly, x = 1/e is where the global minimum occurs for the function
Since e is an irrational number (see proof that e is irrational), it cannot be represented as the quotient of two integers, but it can be represented as a continued fraction. Using calculus, e may also be represented as an infinite series, infinite product, or other types of limit of a sequence.
e aX e bX = e (a + b)X; e X e −X = I; Using the above results, we can easily verify the following claims. If X is symmetric then e X is also symmetric, and if X is skew-symmetric then e X is orthogonal. If X is Hermitian then e X is also Hermitian, and if X is skew-Hermitian then e X is unitary.
In this setting, e 0 = 1, and e x is invertible with inverse e −x for any x in B. If xy = yx, then e x + y = e x e y, but this identity can fail for noncommuting x and y. Some alternative definitions lead to the same function. For instance, e x can be defined as (+).
Consider the roll of a fair die and let A = 1 if the number is even (i.e., 2, 4, or 6) and A = 0 otherwise. ... A conditional expectation of X given ...
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Define e x as the value of the infinite series = =! = + +! +! +! + (Here n! denotes the factorial of n. One proof that e is irrational uses a special case of this formula.) Inverse of logarithm integral.