# Weak Law of Large Numbers

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## 1 Description

The weak law of large numbers is a result in probability theory also known as Bernoulli’s theorem. According to the law, the mean of the results obtained from a large number of trials is close to the population mean.

Let be a sequence of independent and identically distributed random variables, each having a mean and variance .

Define a new variable,

Then,

By the Chebyshev inequality,

In brief,

as , the sample mean equals the population mean .

## 2 Simulation in R

The following is the results of simulations(Bi(n,p)).

Moreover, parameter of the population mean is 0.4, sample number is 1,000.

## 3 Appendix

This is the sample script of R.

Let’s try the Simulation in R with different parameters.

#setting a parameters of Bi(n, p) n <- 1000 p <- 0.4 #dataframe df <- data.frame(bi = rbinom(n, 1, p) ,count = 0, mean = 0) ifelse(df$bi[1] == 1, df[1, 2:3] <- 1, 0) for (i in 2 : n){ df$count[i] <- ifelse(df$bi[i] == 1, df$count[i]<-df$count[i - 1]+1, df$count[i - 1]) df$mean[i] <- df$count[i] / i } #graph plot(df$mean, type='l', main = "Simulation of the Low of Large Numbers", xlab="Numbers", ylab="Sample mean") abline(h = p, col="red")

To

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