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Variable binding relates three things: a variable v, a location a for that variable in an expression and a non-leaf node n of the form Q(v, P). Note: we define a location in an expression as a leaf node in the syntax tree. Variable binding occurs when that location is below the node n. In the lambda calculus, x is a bound variable in the term M ...
Chernoff bounds may also be applied to general sums of independent, bounded random variables, regardless of their distribution; this is known as Hoeffding's inequality. The proof follows a similar approach to the other Chernoff bounds, but applying Hoeffding's lemma to bound the moment generating functions (see Hoeffding's inequality).
Probability bounds analysis (PBA) is a collection of methods of uncertainty propagation for making qualitative and quantitative calculations in the face of uncertainties of various kinds. It is used to project partial information about random variables and other quantities through mathematical expressions.
A random variable is said to be truncated from below if, for some threshold value , the exact value of is known for all cases >, but unknown for all cases . Similarly, truncation from above means the exact value of y {\displaystyle y} is known in cases where y < c {\displaystyle y<c} , but unknown when y ≥ c {\displaystyle y\geq c} .
In probability theory, concentration inequalities provide mathematical bounds on the probability of a random variable deviating from some value (typically, its expected value). The deviation or other function of the random variable can be thought of as a secondary random variable.
That is, for a random variable , the coefficient of variation of + is equal to the coefficient of variation of only when =. In the above example, Celsius can only be converted to Fahrenheit through a linear transformation of the form a x + b {\displaystyle ax+b} with b ≠ 0 {\displaystyle b\neq 0} , whereas Kelvins can be converted to Rankines ...
Markov's inequality (and other similar inequalities) relate probabilities to expectations, and provide (frequently loose but still useful) bounds for the cumulative distribution function of a random variable. Markov's inequality can also be used to upper bound the expectation of a non-negative random variable in terms of its distribution function.
Probability bounds analysis is used to make arithmetic and logical calculations with p-boxes. An example p-box is shown in the figure at right for an uncertain number x consisting of a left (upper) bound and a right (lower) bound on the probability distribution for x. The bounds are coincident for values of x below 0 and above 24. The bounds ...