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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.
To obtain an upper bound for Pr(X > 0), and thus a lower bound for Pr(X = 0), we first note that since X takes only integer values, Pr(X > 0) = Pr(X ≥ 1). Since X is non-negative we can now apply Markov's inequality to obtain Pr(X ≥ 1) ≤ E[X]. Combining these we have Pr(X > 0) ≤ E[X]; the first moment method is simply the use of this ...
In probability theory, Popoviciu's inequality, named after Tiberiu Popoviciu, is an upper bound on the variance σ 2 of any bounded probability distribution. Let M and m be upper and lower bounds on the values of any random variable with a particular probability distribution.
In statistical process control (SPC), the ¯ and R chart is a type of scheme, popularly known as control chart, used to monitor the mean and range of a normally distributed variables simultaneously, when samples are collected at regular intervals from a business or industrial process. [1]
Chebyshev's inequality then follows by dividing by k 2 σ 2. This proof also shows why the bounds are quite loose in typical cases: the conditional expectation on the event where |X − μ| < kσ is thrown away, and the lower bound of k 2 σ 2 on the event |X − μ| ≥ kσ can be quite poor.
This upper bound is the best for the value of s minimizing the value inside the exponential. This can be done easily by optimizing a quadratic, giving = = (). Writing the above bound for this value of s, we get the desired bound:
The upper bound is sharp: M is always a copula, it corresponds to comonotone random variables. The lower bound is point-wise sharp, in the sense that for fixed u , there is a copula C ~ {\displaystyle {\tilde {C}}} such that C ~ ( u ) = W ( u ) {\displaystyle {\tilde {C}}(u)=W(u)} .
In probability theory, a Chernoff bound is an exponentially decreasing upper bound on the tail of a random variable based on its moment generating function. The minimum of all such exponential bounds forms the Chernoff or Chernoff-Cramér bound , which may decay faster than exponential (e.g. sub-Gaussian ).
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