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# STATSLIDE1 - ELEMENTARY PROBABILITY Summary measures of a...

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ELEMENTARY PROBABILITY Summary measures of a statistical experiment. Let us toss the die 30 times and let us record the value assumed by the random variable at each toss: 1 , 2 , 5 , 3 , . . . , 4 , 6 , 2 , 1 . To summarise this information, we may construct a frequency table: x f r 1 8 8 / 30 2 7 7 / 30 3 5 5 / 30 4 5 5 / 30 6 3 3 / 30 6 2 2 / 30 30 1 Here, f i = frequency, n = f i = sample size, r i = f i n = relative frequency. 1

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SUMMARY MEASURES OF THE DISTRIBUTION In this case, the order in which the numbers occur is of no interest; and, therefore, there is no loss of information in creating this summary. We can describe the outcome of the experiment more economically by calculating various summary statistics. First, there is the mean of the sample ¯ x = x i f i n = x i r i . The variance is a measure of the dispersion of the sample relative to the mean, and it is the average of the squared deviations. It is defined by s 2 = ( x i ¯ x ) 2 f i n = ( x i ¯ x ) 2 r i = ( x 2 i x i ¯ x ¯ xx i + { ¯ x } 2 ) r i = x 2 i r i − { ¯ x } 2 , which follows since ¯ x is a constant that is not amenable to the averaging operation. 2
THE CONCEPT OF PROBABILITY It is tempting to define the probabilities of the various outcomes as the limits of the corre- sponding empirical relative frequencies as the sample size n tends to infinity. We may say that a sequence of numbers { r i ; i = 1 , 2 , 3 , . . . } tends to a limit p if, for every number , be it ever so small, there exists a number n = n ( ), which is a function of , such that | r i p | < for all i > n. We may denote this by writing lim( n → ∞ ) r n = p However, this cannot serve in an definition of probability, since there is always a chance that, by a run of aberrant outcomes, the value of r i will break the bounds of the neighbourhood ( p , p ).

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