What is the Central Limit Theorem?

The central limit theorem refers to a class of theorems that discuss the part and distribution of random variable sequences asymptotically normal in probability theory. This set of theorems is the theoretical basis of mathematical statistics and error analysis, and points out the conditions under which a large number of random variables approximately follow a normal distribution. It is the most important class of theorems in probability theory, and has a wide range of practical applications. In nature and production, some phenomena are affected by many independent random factors. If the impact of each factor is very small, the total impact can be regarded as following a normal distribution. The central limit theorem proves this phenomenon mathematically. The earliest central limit theorem was the focus of discussion. In the Bernoulli test, the number of occurrences of event A is asymptotic to the normal distribution. ^[1]

It is the most important category in probability theory

A/B Central Limit Theorem Application of Central Limit Theorem in A / B Testing

The central limit theorem is the most important type of theorem in probability theory, which supports the calculation formulas and related theories of the T test and hypothesis test related to the confidence interval. Without this theorem, the subsequent derivation formulas are not valid.

In fact, the above two interpretations of the central limit theorem can play a role in determining the confidence interval of the index of the A / B test in different scenarios.

For the index data that belongs to the normal distribution, we can quickly perform the next hypothesis test on it and calculate the corresponding confidence interval. For data that do not belong to the normal distribution, according to the central limit theorem, When it is large, the sampling distribution of the overall parameter tends to be normally distributed. In the end, it can be analyzed in the next step according to the normal distribution test formula.

Other examples of central limit theorem

1. A certain artillery position fires 100 times on the enemy's defensive position. The number of hits of the shells in each shot is a random variable. The expectation is 2, and the variance is 1.69. The probability of hitting the target.

Solution: Let Xk be the number of shells in the kth shot, then E (X _i ) = 2, D (X _i ) = 1.69, and S ₁₀₀ = X ₁ + X ₂ + + X ₁₀₀ , and apply the central limit theorem ,

Obeys N (0,1) approximately.

,and so:

Therefore, the probability that 180 to 220 shells will hit the target in 100 shots is 87.64%. ^[4]

2. A complex system consists of 100 independent components. The probability of each component being damaged during system operation is 0.1. In order for the system to work properly, at least 85 components must work. Find the reliability of the system (for normal operation) Probability).

Solution: X is the number of components that work normally among 100 components, then X B (100,0.9), which is approximated by the normal distribution of the binomial distribution,

That is, the probability of normal work is 95.25%. ^[4]