S_Local_Limit_Theorem {ExpRep}

Simulations of Local 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 exactly m times (0<=m<=n). It also calculates the difference between these probabilities and shows all the computations in a table.

Usage

S_Local_Limit_Theorem(n = 170, p = 0.5, Compare = TRUE, Table = TRUE, 
     Graph = TRUE, GraphE = TRUE)

Arguments

n

An integer value representing the number of repetitions of the experiment.

p

A real value with the probability that a successful event will happen in any single Bernoulli experiment (called the probability of success).

Compare

A logical value, if TRUE the function calculates both the exact probability and the approximate probability that a successful event occurs exactly m times 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 exactly m times.

GraphE

A logical value, if TRUE the function shows the graphic of the errors in the approximation.

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. The value of n must be high and the value of p must be small.

Value

A graph and/or a table.

Note

Department of Mathematics. University of Oriente. Cuba.

Author(s)

Larisa Zamora and Jorge Diaz

References

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

See Also

Integral_Theorem, Local_Theorem.

Examples

S_Local_Limit_Theorem(n = 170, p = 0.5, Compare = TRUE, Table = TRUE, Graph = TRUE, 
     GraphE = TRUE)

## The function is currently defined as
function (n = 170, p = 0.5, Compare = TRUE, Table = TRUE, Graph = TRUE, GraphE = TRUE) 
  { layout(matrix(1))
    m <- array(0:n)
    x <- numeric()
    PNormal <- numeric()
    a <- n * p
    b <- sqrt(a * (1 - p))
    for (mi in 1:(n + 1)) {
        x[mi] <- (mi - 1 - a)/b
        PNormal[mi] <- dnorm(x[mi], 0, 1)/b
    }
    if (Compare == TRUE) {
        PBin <- numeric()
        for (mi in 1:(n + 1)) PBin[mi] <- dbinom(mi - 1, n, p)
        Dif <- abs(PBin - PNormal)
    }
    if (Graph == TRUE & GraphE == TRUE) {
        layout(matrix(c(1, 1, 2, 2), 2, 2, byrow = TRUE))
    }
    if (Graph == TRUE) {
        mfg <- c(1, 1, 2, 2)
        plot(PNormal, type = "p", main = "The Local Limit Theorem", 
            xlab = "m", ylab = "Probability", col = "red")
        mtext("Local Theorem", line = -1, side = 3, adj = 1, 
            col = "red")
        if (Compare == TRUE) {
            points(PBin, type = "p", col = "blue")
            mtext("Binomial Probability", line = -2, side = 3, 
                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) {
        if (Compare == TRUE) 
            TablaR <- data.frame(m = m, x = x, PBinomial = PBin, 
                TLocal = PNormal, Difference = Dif)
        else TablaR <- data.frame(m = m, x = x, TLocal = PNormal)
        TablaR
    }
  }