S_Integral_Theorem: Simulations of the Integral Theorem of DeMoivre-Laplace.
In ExpRep: Experiment Repetitions

Description Usage Arguments Details Value Note Author(s) References See Also Examples

Given n Bernoulli experiments with success probability p, this function calculates and plots the exact probability and the approximate probability that a successful event occurs between linf+i (0<=i<=lsup-linf-1) and lsup times. It also calculates the difference between these probabilities and shows all the computations in a table.

1 2	S_Integral_Theorem(n = 200, p = 0.5, linf = 0, lsup = 100, Compare = TRUE, Table = TRUE, Graph = TRUE, GraphE = TRUE)

`n`	An integer vector with the numbers of repetitions of the Bernoulli experiment.
`p`	A real value with the probability that a successful event will happen in any single Bernoulli experiment (called the probability of success).
`linf`	An integer vector, of the same length than n, with the minimum numbers of times that the successful event should happen.
`lsup`	An integer vector, of the same length than n, with the maximum numbers of times that the successful event should happen.
`Compare`	A logical value, if TRUE the function calculates both the exact probability and the approximate probability that a successful event occurs and compares these probabilities.
`Table`	A logical value, if TRUE the function shows a table with the carried out computations.
`Graph`	A logical value, if TRUE the function plots both the exact probability and the approximate probability that a successful event occurs.
`GraphE`	A logical value, if TRUE the function shows the graphic of the errors in the approximation.

Bernoulli experiments are sequences of events, in which successive experiments are independent and at each experiment the probability of appearance of a "successful" event (p) remains constant. It is necessary that linf < lsup.

A graph and/or a table.

Department of Mathematics. University of Oriente. Cuba.

Larisa Zamora and Jorge Diaz

Gnedenko, B. V. (1978). The Theory of Probability. Mir Publishers. Moscow.

Integral_Theorem, Local_Theorem.

S_Integral_Theorem (n = 200, p = 0.5, linf = 0, lsup = 100, Compare = TRUE, Table = TRUE, 
    Graph = TRUE, GraphE = TRUE) 

## The function is currently defined as
function (n = 200, p = 0.5, linf = 0, lsup = 100, Compare = TRUE, 
    Table = TRUE, Graph = TRUE, GraphE = TRUE) 
  { Integral_Theorem <- function(n = 100, p = 0.5, linf = 0, 
        lsup = 50) {
        A <- (linf - n * p)/sqrt(n * p * (1 - p))
        B <- (lsup - n * p)/sqrt(n * p * (1 - p))
        P <- pnorm(B) - pnorm(A)
        return(P)
    }
    layout(matrix(1))
    PNormal <- numeric()
    Dif <- numeric()
    PBin <- numeric()
    k <- lsup - linf - 1
    PNormal[1] <- Integral_Theorem(n, p, linf, lsup)
    PBin[1] <- 0
    for (j in linf:lsup) PBin[1] <- PBin[1] + dbinom(j, n, p)
    Dif[1] <- abs(PBin[1] - PNormal[1])
    for (i in 1:k) {
        linf_i <- linf + i
        PNormal[i + 1] <- Integral_Theorem(n, p, linf_i, lsup)
        if (Compare == TRUE) {
            PBin[i + 1] <- 0
            for (j in linf_i:lsup) PBin[i + 1] <- PBin[i + 1] + 
                dbinom(j, n, p)
            Dif[i + 1] <- abs(PBin[i + 1] - PNormal[i + 1])
        }
    }
    if (Graph == TRUE & GraphE == TRUE) {
        layout(matrix(c(1, 1, 2, 2), 2, 2, byrow = TRUE))
    }
    if (Graph == TRUE) {
        ymini <- min(PNormal[k + 1], PBin[k + 1]) - 0.05
        ymaxi <- max(PNormal[1], PBin[1]) + 0.05
        mfg <- c(1, 1, 2, 2)
        plot(PNormal, ylim = c(ymini, ymaxi), type = "l", main = "The Integral Limit Theorem", 
            xlab = "k (linf<=k<=lsup)", ylab = "Probability", 
            col = "red")
        mtext("Integral Theorem", line = -1, side = 1, adj = 1, 
            col = "red")
        if (Compare == TRUE) {
            points(PBin, type = "p", col = "blue")
            mtext("Binomial Probability", line = -2, side = 1, 
                adj = 1, col = "blue")
        }
    }
    if (GraphE == TRUE) {
        mfg <- c(2, 1, 2, 2)
        dmini <- min(Dif) - 0.01
        dmaxi <- max(Dif) + 0.01
        plot(Dif, ylim = c(dmini, dmaxi), type = "b", main = "Errors", 
            xlab = "m", ylab = "Errors", col = "green")
        abline(a = 0, b = 0, col = "red")
    }
    if (Table == TRUE) {
        Ak <- array(1:(k + 1))
        if (Compare == TRUE) 
            TablaR <- data.frame(k = Ak, PBinomial = PBin, T_Integral = PNormal, 
                Difference = Dif)
        else TablaR <- data.frame(K = Ak, T_Integral = PNormal)
        TablaR
    }
  }