# Simulations of Local Theorem of DeMoivre-Laplace 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. ```r 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. ## Returns A graph and/or a table. ## References Gnedenko, B. V. (1978). The Theory of Probability. Mir Publishers. Moscow. ## Author(s) Larisa Zamora and Jorge Diaz ## Note Department of Mathematics. University of Oriente. Cuba. ## See Also Integral_Theorem, Local_Theorem. ## Examples ```r 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 } } ```

S_Local_Limit_Theorem function