IMPLEMENTATION OF PROBABILITY THEORY IN ECONOMICS ON THE EXAMPLE OF THE LAW OF LARGE NUMBERS

Анотація

Probability theory always works with different kind of problems in various spheres of life. In
this abstract we are going to explain the work of Maths in practice. For example, there are two maternity
hospitals, one small and one large in our city. According to the statistical report, it is stated
that 70% of the babies born this year are boys, and 30% are girls. The question is: which hospital do
you think this statistic likely belongs to — the small or the large one? At the end of this article, we
will answer this question and provide a rationale for it.
The Italian mathematician and engineer Gerolamo Cardano, published his book “De Ludo
Aleae”, where he introduces the idea that the accuracy of empirical statistics improves with the
number of trials. In other words, the more experiments conducted, the closer the obtained average
value is to the true one. This marked the beginning of one of the fundamental law in probability
theory — the Law of Large Numbers (LLN). This law allows us to predict the behavior of a random
variable, whether it’s the average height of a person, the probability of a baby being born male or
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female, the average profit a casino makes per round in a given game, the profitability of insurance
terms for a company, and so on (Contributors to Wikimedia projects, 2002a).
Informal Definition of the Law of Large Numbers. The arithmetic mean of a large number
of independent, identically distributed random variables stabilizes as their number increases (Contributors
to Wikimedia projects, 2002b). Let’s look at an example. Take a six-sided die and repeatedly
roll it, calculating the arithmetic mean of the scores obtained from all rolls after each throw.
Roll 1:
Outcome: 6
Arithmetic mean: 61=6
Roll 2:
Outcome: 2
Arithmetic mean: 6+22=4
…
To automate the simulation of our experiment, let’s write a small Python script. We’ll visualize the
results using the Matplotlib library. You can find source code here:
https://github.com/YehorSeniuk/LLNDiceRollsModeling/blob/main/main.py
Figure 1 - LLN for Dice Rolls
Formal Definition of the Law of Large Numbers. To begin with, there are two types of the
Law of Large Numbers: the Weak Law of Large Numbers (WLLN) and the Strong Law of Large
Numbers (SLLN). They differ in terms of the type of convergence and conditions for applicability.
The Weak LLN uses convergence in probability, while the Strong LLN uses almost sure convergence.
Let’s examine each type of convergence separately.
Convergence in Probability. Let’s consider a sequence of random variables (Xi)i=1n
The sequence of random variables converges in probability if and only if, as n approaches
infinity, for any arbitrarily small positive probability, the probability that the absolute deviation of
the arithmetic mean X from the expected value exceeds tends to zero. In other words:
nP(|Xn-|>)=0 (1)
Almost Sure Convergence. Let’s consider a sequence of random variables (Xi)i=1n
The sequence of random variables converges almost surely if and only if, for almost every
possible outcome, the values of the sequence will eventually get arbitrarily close to a certain value
as the number of observations increases. In other words:
P(nXn=)=1 (2)
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Weak Law of Large Numbers. Let’s consider a sequence of random variables (Xi)i=1n
where all random variables:
1. Pairwise independent
2. Have identical distribution
3. Have finite variance
Then convergence in probability holds, that is nP(|Xn-|>)=0.
Strong Law of Large Numbers. Let’s consider a sequence of random variables (Xi)i=1n
where all random variables:
1. Pairwise independent
2. Have identical distribution
3. Have finite variance
4. Satisfies n=1D[Xn]n2< (Kolmogorov theorem)
Then almost sure convergence holds, that is P(nXn=)=1.
It is also important to note that the fulfillment of the Strong Law of Large Numbers implies the fulfillment
of the Weak Law of Large Numbers; however, the reverse is generally not true. This is
because almost sure convergence of a random variable implies its convergence in probability, but
the reverse is not generally true, that is:
P(nXn=)=1nP(|Xn-|>)=0 (3)
Application. Insurance. Suppose we are opening an insurance company. We face the question:
how can we determine whether a particular insurance offer is profitable for the company, considering
the frequency of claims, the cost of insurance, and the payout in the event of a claim in the
long run? To answer this question, let’s recall the Law of Large Numbers. Thus, if the random variable
that describes the insurance claim meets the criteria of the Law of Large Numbers, we can expect
that empirical statistics, such as the arithmetic mean of income and expenses, will tend toward
its true value, that is, its expected value. Now, all we need to do is calculate this expected value
based on the probability of a claim occurring (obtained from statistics).
Let the random variable Xi represent the income or expense in the i-th insurance case, then
=1ni=1nXiP(X=Xi) (4)
The Promised Answer. Most likely, this statistic belongs to the smaller hospital, and here's
why: it is a statistical fact that boys and girls are born with approximately equal probability, that is,
P(X=M)P(X=G)0.5. The report indicated that 70% of those born are boys, which is a significant
deviation (we would even call it a leap) from the overall statistics; this, in turn, is characteristic of
small samples. Indeed, let’s recall how the arithmetic mean behaves when calculating scores during
dice rolling.
The Law of Large Numbers is one of the fundamental laws of probability theory. It has established
a connection between the number of experiments and the accuracy of the obtained values,
thereby making an invaluable contribution to the experimental sciences, such as physics, astronomy,
medicine, and many others. It is also worth noting its significant impact on the development of
fields such as economic sciences, insurance sphere, illegal gaming and others.