Theory of Probability by Bruno de Finetti

Author:Bruno de Finetti [Finetti, Bruno de]
Language: eng
Format: epub
ISBN: 9781119286295
Publisher: Wiley
Published: 2017-03-20T00:00:00+00:00

7.6 The ‘Central Limit Theorem’; The Normal Distribution

7.6.1. If one draws the histograms of the distribution of Heads and Tails (the binomial distribution with ) and compares them for various values of n (the number of tosses), one sees that the shape remains the same (apart from discontinuities and truncation of the tails, features which arise because of the discreteness, and tend to vanish as n increases). The shape, in fact, suggests that one is dealing with the familiar normal distribution (the Gaussian distribution, or ‘distribution of errors’, which we mentioned briefly in Chapter 6, 6.11.3, and will further treat in Section 7.6.6; see Figure 7.6). Figure 7.5 gives the histogram for n = 9 (which is, in fact, a very small n!), together with the density curve. The agreement is already quite good, and the curve and the boundary of the histogram would rapidly become indistinguishable if we took a larger n (not necessarily very large).

Figure 7.5 The binomial distribution: Heads and Tails with n tosses, n = 9. The possible values for the gain run through the ten odd numbers from −9 to +9, and the height of the column indicates the probability concentrated on each of these numbers. To give a more expressive picture, the values are assigned uniformly within ±1 of each point; this makes much clearer the approach of the binomial to the normal distribution, which will, in fact, be shown to be the limit distribution (as n → ∞).

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