Appendix C An Elementary Derivation of the One-Dimensional Central Limit Theorem from the Random Walk

Fig.

Table C.1 The probability distribution of the one-dimensional random walk

It then follows that:

C.1 The Derivation of the Density Function of the Normal Distribution^{[87] * *}

In the following, we will derive the density function of the normal distribution. We give a Taylor expansion around the maximum k = v on the function ln ∏(k), for convenience of calculation, instead of ∏(k).

b₂ becomes negative, and hence may be rewritten as:

We then assume that λ = k — v is small enough to ignore λ³, ∙ ∙ ∙,λ^k, and we may then reach the next approximation:

We note that:

It then by the famous integral formula on a Gaussian function holds that:

And then:

We examine each item ∏(k).

Since:

It then holds:

We denote E(∆k²) by σ². This creates the next formula for the normal distribution:

This is the density function of the normal distribution.

If p = 0.5 in the last equation, it holds that:

By σ² = E((∆k)²) = N, it also holds that:

QED

Fig. C.2 The probability of long leads

C.2 A Heuristic Finding in the Random Walk

We finally give an interesting result in the random walk. Even now, the research on random walk is under development. As we showed in Section 6.1.1.1, we can find self-similarity in the random walk. We moreover show a heuristic finding of the probability of long leads in a random walk.^[88] Consider a game in which a fair coin is tossed repeatedly. When the cumulative number of heads is greater than the cumulative number of tails, heads is in the lead. It retains that position until the cumulative number of tails is greater. The probability distribution indicates the probability of one side being in the lead for different percentages of the duration of the game. For example, if the coin is tossed 20 times, the probability that heads will be in the lead during the entire course of the game is 0.176, the same as the probability that it will never be in the lead.

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