create_matrix_triangular_simulate_Normal_Restricted.rmd

---
title: "create_matrix_triangular_simulate_Normal_Restricted"
output: github_document
---

```{r setup, include=FALSE}
knitr::opts_chunk$set(echo = TRUE)
Sys.setenv(RSTUDIO_PANDOC = "C:/Program Files/RStudio/bin/quarto/bin/tools")
```

## Ex I.

#### A function that takes a square matrix and sets it to zero above the main diagonal

```{r}

fun_tri_mat <- function(A){
    D <- dim(A)
    if(D[1] != D[2]) stop("A must be square matrix")
    n <- D[1]
    for(i in 1:(n-1)){
        for(j in (i+1):n){
            A[i, j] <- 0
        }
    }
    return(A)
}
A <- matrix(1, 4, 4)
A
A2 <- fun_tri_mat(A)
A2
```



***



## EX II.

#### A function that is a normal sample of the given limits, in such a way that the simulated 
#### values must be larger than the specified value of A and smaller than the specified value of B. 
#### In this function, the size of the production sample and the user's desired mean and standard deviation, 
#### as well as the user's desired limits, are given in the function arguments.



```{r}
create_normal_restrict <- function(size, 
Mu, Sig, a, b){
    i = 0
    Sample <- c()
    while(i <= size){
        temp1 <- rnorm(1, Mu, Sig)
        if(temp1 < b && temp1 > a){
            i = i + 1; Sample[i] <- temp1
        }
    }
    return(Sample)
}
s <- create_normal_restrict(size = 10, Mu = 0, Sig = 2, a = 2, b = 4)
s
all(s > 2) && all(s < 4)

```



***


## EX III. 

#### Edit code for EX II.

```{r}
create_normal_restrict <- function(size, 
Mu, Sig, a, b){
    Sample <- c()
    for(i in 1:size){
        temp1 <- rnorm(1, Mu, Sig)
        while(temp1 < a || temp1 > b){
            temp1 <- rnorm(1, Mu, Sig)
        }
        Sample[i] <- temp1
    }
    return(Sample)
}
s <- create_normal_restrict(size = 20, Mu = 2, Sig = 3, a = 3, b = 5)
s
all(s > 3) && all(s < 5)

```



***


## EX IV. 

#### Simulation to prove the weak law of large numbers

```{r}

Weak_law_large_num <- function(size, Mu, showPlot = TRUE){
    n <- size
    E <- list()
    for(i in 1:n){
        E[[i]] <- rnorm(i, mean = Mu, sd = 1)
    }
    E_mean <- unlist(lapply(E, mean))
    list_result <- list(Mean_result = E_mean, list_result = E)
    if(showPlot){
        plot(x = 1:n, y = E_mean, col = "darkblue", type = "l", xlab = "size of sample", ylab = "Sample Mean")
        abline(h = Mu, lwd = 1.5, lty = 2, col = "yellow")
        legend("topright", legend = expression(mu), lty = 2, lwd = 3, col = "yellow", bg = "black", cex = 3, 
        text.col = "white")
    }
    return(list_result)
}

Result <- Weak_law_large_num(size = 1e+4, Mu = 2.5, showPlot = T) 
E <- Result$list_result
E[[5]]
E[[10]]
E[[20]]
E[[50]]
```