ExpRep (version 1.0)

S_Integral_Theorem: Simulations of the Integral Theorem of DeMoivre-Laplace.

Description

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.

Usage

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

Arguments

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.

Value

A graph and/or a table.

Details

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.

References

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

See Also

Integral_Theorem, Local_Theorem.

Examples

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
    }
  }