It has been recognised that this is an inappropriate

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It has been recognised that this is an inappropriate way of defining probabil- ities. We cannot regard such a limit of relative frequency in the same light as an ordinary mathematical limit. We cannot say, as we can for a mathematical limit, that, when n exceeds a certain number, the relative frequency r i is bound there- after to reside within a limited neighbourhood of p i . (Such a neighbourhood is denoted by the interval ( p + n , p n ), where n is a number that gets smaller as n increases.) The reason is that there is always a wayward chance that, by a run of aberrant outcomes, the value of r i will break the bounds of the neighbourhood. All that we can say is that the probability of its doing so becomes vanishingly small as n → ∞ . But, in making this statement, we have invoked a notion of probability, which is the very thing that we have intended to define. Clearly, such a limiting process cannot serve to define probabilities. We shall continue, for the present, to assume that we understand unquestion- ably that the probability of any number x i arising from the toss of a fair die is p i = 1 / 6 for all i . Indeed, this constitutes our definition of a fair die. We can replace all of the relative frequencies in the formulae above by the corresponding probabilities to derive a new set of measures that characterise the idealised notion of the long run tendencies of these statistics. Population measures. We define the mathematical expectation or mean of the random variable associated with a toss of the die to be the number E ( x ) = µ = x i p i = 1 6 (1 + 2 + 3 + 4 + 5 + 6) = 21 6 = 3 . 5 . 2
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Likewise, we may define V ( x ) = σ 2 = ( { x i E ( x ) } 2 p i = ( x 2 i x i E ( x ) E ( x ) x i + { E ( x ) } 2 ) p i = x 2 i p i − { E ( x ) } 2 , which we may call the true variance of the population variance. We may calculate that V ( x ) = 13 . 125 . Random sampling. By calling this the population variance, we are invoking a concept of an ideal population in which the relative frequencies are precisely equal to the probabilities that we have defined. In that case, the sample experiment that we described at the beginning is a matter of selecting elements of the population at random, of recording their values and, thereafter, of returning them to the pop- ulation to take their place amongst the other elements that have an equal chance of being selected. In effect, we are providing a definition of random sampling. The union of mutually exclusive events. Having settled our understanding of probability, at least within the present context, let us return to the business of defining compound events. We might ask ourselves, what is the probability of getting either a 2 or a 4 or a 6 in a single toss of the die? We can denote the event in question by A 2 A 4 A 6 .
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  • Spring '12
  • D.S.G.Pollock
  • Probability, Probability theory, Ri, σ -field

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