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An event, however, is any subset of the sample space, including any singleton set (an elementary event), the empty set (an impossible event, with probability zero) and the sample space itself (a certain event, with probability one). Other events are proper subsets of the sample space that contain multiple elements. So, for example, potential ...
In probability experiments on a finite sample space with a non-zero probability for each outcome, there is no difference between almost surely and surely (since having a probability of 1 entails including all the sample points); however, this distinction becomes important when the sample space is an infinite set, [2] because an infinite set can ...
In probability theory, an elementary event, also called an atomic event or sample point, is an event which contains only a single outcome in the sample space. [1] Using set theory terminology, an elementary event is a singleton. Elementary events and their corresponding outcomes are often written interchangeably for simplicity, as such an event ...
If X n converges in probability to X, and if P(| X n | ≤ b) = 1 for all n and some b, then X n converges in rth mean to X for all r ≥ 1. In other words, if X n converges in probability to X and all random variables X n are almost surely bounded above and below, then X n converges to X also in any rth mean. [10] Almost sure representation ...
For example, the probability of the union of the mutually exclusive events and in the random experiment of one coin toss, (), is the sum of probability for and the probability for , () + (). Second, the probability of the sample space Ω {\displaystyle \Omega } must be equal to 1 (which accounts for the fact that, given an execution of the ...
Tail events are precisely those events whose occurrence can still be determined if an arbitrarily large but finite initial segment of the is removed. In many situations, it can be easy to apply Kolmogorov's zero–one law to show that some event has probability 0 or 1, but surprisingly hard to determine which of these two extreme values is the ...
Each of the probabilities on the right-hand side converge to zero as n → ∞ by definition of the convergence of {X n} and {Y n} in probability to X and Y respectively. Taking the limit we conclude that the left-hand side also converges to zero, and therefore the sequence {(X n, Y n)} converges in probability to {(X, Y)}.
The graph on the right shows the probability density function of r given that 7 heads were obtained in 10 tosses. (Note: r is the probability of obtaining heads when tossing the same coin once.) Plot of the probability density f(r | H = 7, T = 3) = 1320 r 7 (1 − r) 3 with r ranging from 0 to 1