Law of Large Numbers: Statistical Expectation and Predictive Accuracy

The Law of Large Numbers states that the greater the number of exposures, the more accurate the prediction of outcomes, less deviation from expected losses, and greater credibility of the prediction, a foundation for calculating insurance premiums.

The Law of Large Numbers (LLN) is a fundamental theorem in probability and statistics which asserts that as the size of a sample increases, the average of the sample values (mean) becomes increasingly close to the expected value. LLN underpins many practical applications, particularly in fields like insurance, finance, and risk management.

Mathematical Premise

Definition and Formula

The Law of Large Numbers can be mathematically defined as:

$$ \lim_{n \to \infty} \frac{1}{n} \sum_{i=1}^{n} X_i = \mu $$

where \( X_i \) are independent, identically distributed random variables with expected value \( \mu \). As \( n \) approaches infinity, the sample mean \( \frac{1}{n} \sum_{i=1}^{n} X_i \) converges to the expected value \( \mu \).

Types of Law of Large Numbers

Weak Law of Large Numbers (WLLN)

The WLLN states that for a sufficiently large sample size, the sample mean will be close to the expected value in probability, i.e., \( \overline{X}_n \) converges in probability to \( \mu \).

Strong Law of Large Numbers (SLLN)

The SLLN states that the sample mean almost surely converges to the expected value as the number of trials approaches infinity. This almost sure convergence implies a stronger form of reliability in practical applications.

Applications in Insurance

Premium Calculation

The LLN forms the basis for calculating insurance premiums. Insurance companies rely on the principle that as the number of policyholders increases, the actual loss experience will converge to the expected losses, making the prediction of losses and setting of premiums more accurate.

Credibility Theory

In insurance, credibility refers to the degree of confidence in the prediction of future losses. As the number of exposures increases, credibility approaches one, meaning that the prediction is highly reliable.

Examples and Illustrations

Coin Toss Example

Consider the simple case of tossing a fair coin. The expected value for heads in a single toss is 0.5. As the number of coin tosses increases (e.g., 1,000 or 10,000), the proportion of heads will converge closer to 0.5.

Insurance Example

An insurance company predicts that 2% of policyholders will file a claim. If only 100 policies are sold, the variance from this prediction can be high. However, if 10,000 policies are sold, the actual percentage of claims will likely be very close to the predicted 2%.

Historical Context

The Law of Large Numbers was first formulated by the Swiss mathematician Jakob Bernoulli in the late 17th century and published posthumously in his work “Ars Conjectandi” in 1713. Bernoulli’s insight laid the groundwork for the development of modern probability theory and statistical inference.

Key Considerations

Independence and Distribution

For the LLN to hold, the random variables involved must be independent and identically distributed. Violations of these conditions can compromise the reliability of the results.

Practical Limits

While the LLN indicates convergence with large samples, it does not specify the number of trials required for a ’large’ sample, which depends on the variance of the underlying distribution.

Comparisons with Related Terms

Central Limit Theorem (CLT)

The CLT stipulates that the distribution of the sample mean will approach a normal distribution as the sample size grows, regardless of the original population distribution.

Law of Averages

The Law of Averages is a common misunderstanding that implies outcomes of random events will “even out” in the short term, which LLN does not support.

FAQs

How is the Law of Large Numbers used in finance?

In finance, the LLN helps in diversification whereby the risk of a portfolio is reduced as the number of assets in the portfolio increases. This is because the average return of a large number of assets tends to stabilize.

Is the Law of Large Numbers applicable to small sample sizes?

No, the LLN specifically applies to large sample sizes. Small samples may still exhibit significant deviation from the expected value due to higher variance.

Can the Law of Large Numbers be used to predict individual outcomes?

No, LLN cannot predict individual outcomes but rather the average outcome over a large number of trials.

References

• Bernoulli, J. (1713). Ars Conjectandi. Basel.
• Ross, S. (2010). A First Course in Probability. Pearson.
• Feller, W. (1968). An Introduction to Probability Theory and Its Applications. Wiley.

Summary

The Law of Large Numbers is a pivotal concept in probability and statistics, emphasizing that as the number of trials or exposures increases, the average of the outcomes becomes more predictable and converges to the expected value. This principle is extensively used in various domains including insurance, finance, and risk management to ensure accurate predictions and risk assessments.