create_matrix_triangular_simulate_Normal_Restricted.md

create_matrix_triangular_simulate_Normal_Restricted

Ex I.

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

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

##      [,1] [,2] [,3] [,4]
## [1,]    1    1    1    1
## [2,]    1    1    1    1
## [3,]    1    1    1    1
## [4,]    1    1    1    1

A2 <- fun_tri_mat(A)
A2

##      [,1] [,2] [,3] [,4]
## [1,]    1    0    0    0
## [2,]    1    1    0    0
## [3,]    1    1    1    0
## [4,]    1    1    1    1

---

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.

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

##  [1] 2.243753 2.141859 2.499287 2.411613 2.786077 3.007454 2.016532 2.194721
##  [9] 2.960471 3.566795 3.004377

all(s > 2) && all(s < 4)

## [1] TRUE

---

EX III.

Edit code for EX II.

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

##  [1] 3.911585 3.921723 3.122110 3.948795 4.774382 3.076863 3.469231 4.365678
##  [9] 3.269514 4.202419 3.463050 3.377031 3.380355 3.799234 4.243608 4.837978
## [17] 3.189766 4.140438 3.525532 3.183216

all(s > 3) && all(s < 5)

## [1] TRUE

---

EX IV.

Simulation to prove the weak law of large numbers

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]]

## [1] 3.116940 4.857782 1.882710 1.226727 1.481583

E[[10]]

##  [1] 3.292993 2.268598 3.157134 3.571759 3.159709 1.713299 4.043080 3.037837
##  [9] 2.692407 1.889819

E[[20]]

##  [1] -0.6664324  4.8782451  2.9060415  0.6526347  1.9626721  0.7180531
##  [7]  3.2976689  3.4396635  1.9731405  2.4000012  1.4585400  3.0614244
## [13]  2.0365139  3.8850386  2.3145039  3.2419005  4.0532249  2.6540564
## [19]  2.6158590  3.2020568

E[[50]]

##  [1] 2.0276034 2.1878928 4.3126485 2.5441813 0.3146525 2.4537864 3.5692615
##  [8] 3.7117593 0.9467995 2.6343303 0.5337894 2.4028012 1.1944789 2.2382669
## [15] 1.6622360 2.8248932 3.2131675 3.0547236 3.0619314 1.9556758 2.4052841
## [22] 2.4501912 1.2803878 2.2348861 1.8950170 2.2464262 0.9141530 2.1164518
## [29] 3.2233743 1.5031934 5.0491144 2.7211629 2.6247484 2.4374962 0.8734581
## [36] 2.7386009 1.3735067 1.2746713 0.8909878 2.0065375 2.0193940 0.4261524
## [43] 2.3482436 2.7726120 2.3382846 2.5743570 3.2558597 0.8673530 3.0893295
## [50] 1.3701027