 |
Business | Industries | Finance | Tax |
| Index: > A B C D E F G H I J K L M N O P Q R S T U V W X Y Z |
|
|||||
| First Prev [ 1 2 ] Next Last |
Two random variables X and Y are equal in p-th mean if the pth moment of |X − Y| is zero, that is,
Equality in pth mean implies equality in qth mean for all q<p. As in the previous case, there is a related distance between the random variables, namely
Two random variables X and Y are equal almost surely if, and only if, the probability that they are different is zero:
For all practical purposes in probability theory, this notion of equivalence is as strong as actual equality. It is associated to the following distance:
where 'sup' in this case represents the essential supremum in the sense of measure theory.
Finally, two random variables X and Y are equal if they are equal as functions on their probability space, that is,
Much of mathematical statistics consists in proving convergence results for certain sequences of random variables; see for instance the law of large numbersIn a statistical context, laws of large numbers imply that the average of a random sample from a large population is likely to be close to the mean of the whole population. In probability theory, several laws of large numbers say that the average of a seq and the central limit theoremCentral limit theorems are a set of weak-convergence results in probability theory. Intuitively, they all express the fact that any sum of many independent identically distributed random variables is approximately normally distributed. These results expla.
There are various senses in which a sequence (Xn) of random variables can converge to a random variable X. These are explained in the article on convergence of random variablesIn probability theory, there exist several different notions of convergence of random variables. The convergence (in one of the senses presented below) of sequences of random variables to some limiting random variable is an important concept in probabilit.
The following are examples of random integers i, 1 ≤ i ≤ 100:
17 12 17 89 64 4 62 6 82 80 61 100 19 7 35 4 23 43 49 69 4 81 64 52 33 59 56 56 46 25 2 44 34 73 58 48 94 18 65 47 73 16 69 26 26 65 35 65 64 2 59 36 52 77 52 14 79 42 71 82 60 28 72 96 77 72 78 58 71 44 99 41 41 80 53 67 7 66 49 86 94 85 47 27 1 6 86 50 32 26 60 79 94 53 72 98 78 46 73 50 49 3 77 57 56 23 20 70 1 58 42 72 16 84 96 44 42 76 19 71 57 17 34 66 68 63 100 37 38 68 52 52 42 86 15 53 76 59 43 94 67 21 74 73 85 16 12 45 57 7 4 22 23 74 15 63 80 65 76 88 39 39 100 96 85 64 16 55 62 50 71 27 48 95 96 30 65 33 71 50 39 1 70 99 55 74 2 74 98 48 99 90 28 66 41 17 80 35 8 30 85 41 68 18 46 86 91 40 20 43 71 95 48 40 79 88 77 49 81 52 15 8 11 51 26 99 8 28 37 47 37 17 30 27 39 33 65 8 31 73 48 96 41 78 9 89 72 16 61 48 73 90 39 34 7 41 1 87 48 83 41 64 61 47 71 2 35 66 74 29 74 7 61 22 46 46 4 59 23 79 33 7 31 41 54 63 91 81 58 66 83 24 37 84 16 55 9 52 92 69 44 27 57 38 70 37 33 23 24 18 74 20 87 73 28 85 34 31 76 25 6 38 15 73 16 79 83 94 21 52 34 19 66 5 97 33 100 63 36 100 4 63 84 8 21 21 92 60 72 22 25 80 23 8 10 10 63 44 14 86 47 17 45 4 18 21 44 27 88 10 92 90 27 54 73 68 13 15 68 31 4 83 46 97 97 32 12 66 66 87 100 75 99 75 73 16 86 90 66 51 59 80 87 40 35 21 76 65 74 73 26 41 17 67 88 54 42 62 98 78 19 29 60 79 19 76 13 95 68 76 86 47 91 23 25 50 57 27 97 30 16 82 5 7 31 72 64 18 32 100 54 18 51 66 38 74 91 75 41 81 21 32 96 78 90 9 82 21 84 80 65 72 52 17 81 50 1 90 14 45 11 76 91 31 20 93 30 30 66 10 20 37 89 3 71 35 96 82 11 4
| Politics | Business | Finance |
 |
 |
 |
|