Experiment Repetitions

Description

The package ExpRep, which basically responds to educational purposes, allows to calculate the probabilities of occurrences of an event in a great number of repetitions of Bernoulli experiment, through the application of the local and the integral theorem of De Moivre Laplace, and the theorem of Poisson. It gives the possibility to show the results graphically and analytically, and to compare the results obtained by the application of the above theorems with those calculated by the direct application of the Binomial formula.

Details

The DESCRIPTION file:

Index of help topics:

ApplicIntegralTheo      Applications of the Integral Theorem of
                        DeMoivre-Laplace.
Buffon                  Buffon
ExpRep-package          Experiment Repetitions
Integral_Theorem        Integral Theorem of DeMoivre-Laplace
Local_Theorem           Local Theorem of DeMoivre-Laplace
Poisson_Theorem         Poisson Theorem.
S_Integral_Theorem      Simulations of the Integral Theorem of
                        DeMoivre-Laplace.
S_Local_Limit_Theorem   Simulations of Local Theorem of
                        DeMoivre-Laplace
S_Poisson_Theorem       Simulations of Poisson Theorem

Author(s)

Larisa Zamora-Matamoros and Jorge Diaz-Silvera

Maintainer: Larisa Zamora-Matamoros <larisa@uo.edu.cu>

References

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

Examples

ProbL<-Local_Theorem(n=100,m=50,p=0.02)
ProbL

ProbI<-Integral_Theorem(n=100,p=0.5,linf=0,lsup=50)
ProbI

ProbP<-Poisson_Theorem(n=100,m=50,p=0.002)
ProbP

beta<-ApplicIntegralTheo(Applic="beta",n=369,p=0.4,alpha=0.05) 
beta

alpha<-ApplicIntegralTheo(Applic="alpha",n=369,p=0.4,beta=0.95) 
alpha

n<-ApplicIntegralTheo(Applic="n",p=0.4,alpha=0.05,beta=0.95) 
n

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

S_Poisson_Theorem(n = 169, p = 0.002, Compare = TRUE, Table = TRUE, Graph = TRUE, 
      GraphE = TRUE)

S_Integral_Theorem(n=100, p=0.5, linf = 0, lsup = 50, Compare = TRUE, Table = TRUE, 
      Graph = TRUE, GraphE = TRUE) 

Buffon(p = 0.5, width = 0.2, r = c(100, 500, 1000, 1500))

Applications of the Integral Theorem of DeMoivre-Laplace.

Description

This function shows three applications of the integral theorem of DeMoivre-Laplace: 1. To estimate the probability (beta) that the frequency of occurrence of the successful event will deviate from the probability that this event will happen in any single Bernoulli experiment (p) in a quantity not bigger than alpha. 2. To calculate the least number of experiments that must be carried out (n). 3. To determine the boundary of possible variations between the frequency of occurrence of the successful event and the probability p (alpha).

Usage

ApplicIntegralTheo(Applic = "alpha", n = 10000, p = 0.5, alpha = 0.01, beta = 0.9)

Arguments

Value

Numeric value representing the values of n, alpha or beta according to the value that the parameter "Applic" takes ("n", "alpha" or "beta").

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.

Examples

beta<-ApplicIntegralTheo(Applic="beta",n=369,p=0.4,alpha=0.05) 
beta

alpha<-ApplicIntegralTheo(Applic="alpha",n=369,p=0.4,beta=0.95) 
alpha

n<-ApplicIntegralTheo(Applic="n",p=0.4,alpha=0.05,beta=0.95) 
n

## The function is currently defined as
function (Applic = "alpha", n = 10000, p = 0.5, alpha = 0.01, 
    beta = 0.9) 
{
    Alpha <- function(n, p, beta) {
        a <- (beta + 1)/2
        alpha <- ((p * (1 - p)/n)^0.5) * qnorm(a)
        return(alpha)
    }
    Beta <- function(n, p, alpha) {
        b <- alpha * (n/(p * (1 - p)))^0.5
        beta <- 2 * pnorm(b) - 1
        return(beta)
    }
    Repetitions <- function(p, alpha, beta) {
        a <- (beta + 1)/2
        n <- (p * (1 - p) * ((qnorm(a)/alpha)^2))%/%1 + 1
        return(n)
    }
    options(digits = 17)
    value <- switch(Applic, alpha = Alpha(n, p, beta), beta = Beta(n, 
        p, alpha), n = Repetitions(p, alpha, beta))
    return(value)
  }

Buffon

Description

Simulations of the experiment of Buffon.

Usage

Buffon(p = 0.5, width = 0.2, r = c(100, 500, 1000, 1500))

Arguments

Value

Four graphics, each one is the simulation of the experiment of Buffon for the number of repetitions contained in the array r.

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.

Examples

Buffon(p = 0.5, width = 0.2, r = c(100, 500, 1000, 1500))

## The function is currently defined as
function (p = 0.5, width = 0.2, r = c(100, 500, 1000, 1500)) 
{
    Position <- function(k, colum) {
        PE <- k%/%colum
        Resto <- k%%colum
        if (Resto == 0) {
            fila <- PE
            columna <- colum
        }
        else {
            fila <- PE + 1
            columna <- Resto
        }
        Position <- list(fila, columna)
        return(Position)
    }
    nf <- layout(matrix(c(1, 2, 3, 4), 2, 2, byrow = TRUE), TRUE)
    k <- 0
    la <- p - width
    lb <- p + width
    if (la < 0) 
        la <- 0
    if (lb > 1) 
        lb <- 1
    for (j in 1:4) {
        k <- k + 1
        Probcara <- array(0, dim = r[j])
        for (i in 1:r[j]) {
            binomial <- rbinom(i, 1, p)
            cara <- length(binomial[binomial == 1])
            Probcara[i] <- cara/i
        }
        P <- Position(k, 2)
        fila <- P[[1]]
        colum <- P[[2]]
        mfg <- c(fila, colum, 2, 2)
        a <- as.character(r[j])
        plot(Probcara, type = "p", main = paste0("n=", a), xlab = "Repetitions", 
            ylab = "Probability", font.main = 3, col = "blue", 
            ylim = c(la, lb))
        abline(h = p, col = "red", lty = 1, lwd = 2)
    }
  }

Integral Theorem of DeMoivre-Laplace

Description

Given n Bernoulli experiments, with success probability p, this function calculates the probability that a successful event occurs between linf and lsup times.

Usage

Integral_Theorem(n = 100, p = 0.5, linf = 0, lsup = 50)

Arguments

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. It is necessary that linf < lsup.

Value

A real value representing the approximate probability that a successful event occurs between linf and lsup times, in n repetitions of a Bernoulli experiment.

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

Poisson_Theorem, Local_Theorem.

Examples

Prob<-Integral_Theorem(n=100,p=0.5,linf=0,lsup=50)
Prob

## The function is currently defined as
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)
  }

Local Theorem of DeMoivre-Laplace

Description

Given n Bernoulli experiments, with success probability p, this function calculates the approximate probability that a successful event occurs exactly m times.

Usage

Local_Theorem(n, m, p)

Arguments

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 real value representing the approximate probability that a successful event occurs exactly m times in n repetitions of a Bernoulli experiment.

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, Poisson_Theorem.

Examples

Prob<-Local_Theorem(n=100,m=50,p=0.02)
Prob

## The function is currently defined as
function (n, m, p) 
{
    a <- n * p
    b <- sqrt(a * (1 - p))
    x <- (m - a)/b
    P <- dnorm(x, 0, 1)/b
    return(P)
  }

Poisson Theorem.

Description

Given n Bernoulli experiments, with success probability p (p small), this function calculates the approximate probability that a successful event occurs exactly m times.

Usage

Poisson_Theorem(n, m, p)

Arguments

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 very small.

Value

A numerical value representing the approximate probability that a successful event occurs exactly m times.

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

Prob<-Poisson_Theorem(n=100,m=50,p=0.002)
Prob

## The function is currently defined as
function (n, m, p) 
{
    landa <- n * p
    P <- dpois(m, landa)
    return(P)
  }

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

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.

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_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
    }
  }

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

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
    }
  }

Simulations of Poisson Theorem

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 theses probabilities and shows all the computations in a table.

Usage

S_Poisson_Theorem(n = 2000, p = 0.002, Compare = TRUE, Table = TRUE, 
     Graph = TRUE, GraphE = FALSE)

Arguments

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 very 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_Poisson_Theorem(n = 169, p = 0.002, Compare = TRUE, Table = TRUE, Graph = TRUE, 
    GraphE = TRUE)
## The function is currently defined as
function (n = 2000, p = 0.002, Compare = TRUE, Table = TRUE, 
    Graph = TRUE, GraphE = FALSE) 
  { layout(matrix(1))
    m <- array(0:n)
    PPoisson <- numeric()
    a <- n * p
    for (mi in 1:(n + 1)) PPoisson[mi] <- dpois(mi - 1, a)
    if (Compare == TRUE) {
        PBin <- numeric()
        x <- numeric()
        PNormal <- numeric()
        Dif1 <- numeric()
        Dif2 <- numeric()
        b <- sqrt(a * (1 - p))
        for (mi in 1:(n + 1)) {
            x[mi] <- (mi - 1 - a)/b
            PBin[mi] <- dbinom(mi - 1, n, p)
            PNormal[mi] <- dnorm(x[mi], 0, 1)/b
        }
        Dif1 <- abs(PBin - PPoisson)
        Dif2 <- 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)
        ll <- length(which(Dif1 > 5e-07))
        plot(PPoisson[1:ll], type = "b", main = "The Poisson Theorem", 
            xlab = "m", ylab = "Probability", col = "red")
        mtext("Poisson Theorem", line = -1, side = 3, adj = 1, 
            col = "red")
        if (Compare == TRUE) {
            points(PBin[1:ll], type = "b", col = "green")
            points(PNormal[1:ll], type = "b", col = "blue")
            mtext("Local Theorem", line = -2, side = 3, adj = 1, 
                col = "blue")
            mtext("Binomial Probability", line = -3, side = 3, 
                adj = 1, col = "green")
        }
    }
    if (GraphE == TRUE) {
        mfg <- c(2, 1, 2, 2)
        ll <- length(which(Dif1 > 5e-07))
        plot(Dif2[1:ll], type = "b", main = "Errors", xlab = "m", 
            ylab = "Differences", col = "red")
        mtext("Binomial-Poisson", line = -1, side = 3, adj = 1, 
            col = "red")
        points(Dif1[1:ll], type = "b", col = "green")
        mtext("Binomial-Local Theorem", line = -2, side = 3, 
            adj = 1, col = "green")
    }
    if (Table == TRUE) {
        if (Compare == TRUE) 
            TablaR <- data.frame(m = m, x = x, PBinomial = PBin, 
                TPoisson = PPoisson, Difference1 = Dif1, TLocal = PNormal, 
                Difference2 = Dif2)
        else TablaR <- data.frame(m = m, TPoisson = PPoisson)
        TablaR
    }
  }