Illustration of the strong law of large numbers and the weak law of large numbers

The following code is from:

https://www.douban.com/note/618668311/

The following is the R code for the strong law of numbers:

n <- 1000;
m <- 50;
e <- 0.05
s <- cumsum(1*(rbinom(n, size=1, prob=0.5) - 0))
plot(s/seq.int(n)-0.5, type = "l", ylim = c(-0.4, 0.4))
abline(h = c(-e,e), lty = 2)
paste("---------------------")

Ctrl+a to select all the code, then Ctrl+Enter to run it in Rstudio,

operation result:

Code Explanation:

rbinom (n, size, prob) n represents the number of random numbers generated, size represents the number of Berkli tests performed, and prob represents the probability of a successful Berkli test.

ylim refers to the upper and lower limit settings of the y-axis

h (horizontal): Represents a horizontal horizontal line. In the code, it means to draw two horizontal lines horizontally. The function values ​​corresponding to the horizontal lines are e and -e respectively.

lty: line type

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The following is the R code for the weak law of large numbers:

n<-1000
m<-50


x <- matrix(1*(rbinom(n*m, size=1, prob=0.5) - 0.0), ncol = m) 
print(x)


y <- apply(x, 2, function(z)(cumsum(z)/seq_along(z))-0.5)


matplot(y, type = "l", ylim = c(-0.4,0.4))  


abline (h = c (-e, e), lty = 2, lwd = 2)  

where u=0

Where cumsum(z)/seq_along(z) is the entire function body

Do m experiments. Each experiment has n samples.

For the vector a=(a1,...,an), cumsum(a) gets a1,a1+a2,a1+a2+a3,....a1+...+an

lwd is the width of the line

apply(x,2, function(z) cumsum(z)/seq_along(z)) means that each column value in the matrix is ​​substituted into function(z)

Because it obeys Bernoulli distribution

The results are as follows:

The usage reference of apply:

http://blog.fens.me/r-apply/