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Any definition of expected value may be extended to define an expected value of a multidimensional random variable, i.e. a random vector X. It is defined component by component, as E[X] i = E[X i]. Similarly, one may define the expected value of a random matrix X with components X ij by E[X] ij = E[X ij].
Therefore, the expected value of the roll is: + + + + + = According to the law of large numbers, if a large number of six-sided dice are rolled, the average of their values (sometimes called the sample mean) will approach 3.5, with the precision increasing as more dice are rolled.
The problem of estimating the maximum of a discrete uniform distribution on the integer interval [,] from a sample of k observations is commonly known as the German tank problem, following the practical application of this maximum estimation problem, during World War II, by Allied forces seeking to estimate German tank production.
In the dice example the standard deviation is √ 2.9 ≈ 1.7, slightly larger than the expected absolute deviation of 1.5. The standard deviation and the expected absolute deviation can both be used as an indicator of the "spread" of a distribution.
We know the expected value exists. The dice throws are randomly distributed and independent of each other. So simple Monte Carlo is applicable: s = 0; for i = 1 to n do throw the three dice until T is met or first exceeded; r i = the number of throws; s = s + r i; repeat m = s / n; If n is large enough, m will be within ε of μ for any ε > 0.
As the factory is improved, the dice become less and less loaded, and the outcomes from tossing a newly produced die will follow the uniform distribution more and more closely. Tossing coins; Let X n be the fraction of heads after tossing up an unbiased coin n times. Then X 1 has the Bernoulli distribution with expected value μ = 0.5 and ...
The information gain in decision trees (,), which is equal to the difference between the entropy of and the conditional entropy of given , quantifies the expected information, or the reduction in entropy, from additionally knowing the value of an attribute . The information gain is used to identify which attributes of the dataset provide the ...
The Dice-Sørensen coefficient (see below for other names) is a statistic used to gauge the similarity of two samples. It was independently developed by the botanists Lee Raymond Dice [ 1 ] and Thorvald Sørensen, [ 2 ] who published in 1945 and 1948 respectively.