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IntroProbDS/ch06.r
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############# | |
# Chapter 6.2 (Probability Inequality) | |
## Compare Chebyshev's and Chernoff's bounds | |
# R code to compare the probability bounds | |
library(pracma) | |
epsilon <- 0.1 | |
sigma <- 1; | |
N <- logspace(1,3.9,50) | |
p_exact <- 1-pnorm(sqrt(N)*epsilon/sigma) | |
p_cheby <- sigma^2 / (epsilon^2*N) | |
p_chern <- exp(-epsilon^2*N/(2*sigma*2)) | |
plot(log(N), log(p_exact), pch=1, col="orange", lwd=4, xlab="log(N)", ylab="log(Probability)") | |
lines(log(N), log(p_cheby), lty=6, col="green", lwd=4) | |
lines(log(N), log(p_chern), pch=19, col="blue", lwd=4) | |
legend(3, -25, c("Exact","Chebyshev","Chernoff"), fill=c("orange", "green", "blue")) | |
############# | |
# Chapter 6.3 (Law of Large Numbers) | |
## Weak law of large numbers | |
# R code to illustrate the weak law of large numbers | |
library(pracma) | |
p <- 0.5 | |
Nset <- as.integer(round(logspace(2,5,100))) | |
x <- matrix(rep(0, 1000*length(Nset)), nrow=1000) | |
for (i in 1:length(Nset)) { | |
N = Nset[i] | |
x[,i] <- rbinom(1000, N, p) / N | |
} | |
Nset_grid <- repmat(Nset, m=1, n=1000) | |
semilogx(Nset_grid, x, col='black', pch=4) | |
lines(Nset, p + 3*(((p*(1-p))/Nset)^(1/2)), col='red', lwd=6) | |
lines(Nset, p - 3*(((p*(1-p))/Nset)^(1/2)), col='red', lwd=6) | |
############# | |
# Chapter 6.4 (Central Limit Theorem) | |
## PDF of the sum of two Gaussians | |
# Plot the PDF of the sum of two Gaussians | |
library(pracma) | |
n <- 10000 | |
K <- 2 | |
Z <- rep(0, n) | |
for (i in 1:K) { | |
X <- runif(n, min=1, max=6) | |
Z <- Z + X | |
} | |
hist(Z,breaks=(K-0.5):(6*K+0.5),freq=FALSE) | |
# Visualize convergence in distribution | |
N <- 10 | |
N <- 50 | |
x <- seq(0, N, (N/1000)) | |
p <- 0.5 | |
p_b <- dbinom(x, N, p) | |
p_n <- dnorm(x, N*p, (N*p*(1-p))**(1/2)) | |
c_b <- pbinom(x, N, p) | |
c_n <- pnorm(x, N*p, (N*p*(1-p))**(1/2)) | |
plot(x, p_n, lwd=6, type="l", col='red') | |
lines(x, p_b, lwd=2, col='black') | |
legend("topright", c('Binomial', 'Gaussian'), col=c('black', 'red'), lty=c(1, 1), lwd=3) | |
# Poisson to Gaussian: convergence in distribution | |
N <- 4 | |
# N = 10 | |
# N = 50 | |
x <- seq(0, 2*N,(2*N/1000)) | |
lambda <- 1 | |
p_b <- dpois(x, N*lambda) | |
p_n <- dnorm(x, N*lambda, sqrt(N*lambda)) | |
c_b <- ppois(x, N*lambda); | |
c_n = pnorm(x, N*lambda, sqrt(N*lambda)); | |
plot(x, p_b, col="black", type="s") | |
lines(x, p_n, col="red", lwd=6) | |
legend("bottomright", c('Poisson', 'Gaussian'), fill=c('black', 'red')) | |
plot(x, c_b, col="black", type="s") | |
lines(x, c_n, col="red", lwd=6) | |
legend("bottomright", c('Poisson', 'Gaussian'), fill=c('black', 'red')) | |
# Visualize the Central Limit Theorem | |
N <- 10 | |
x <- seq(0, N, length.out=1001) | |
p <- 0.5 | |
p_b <- dbinom(x, N, p); | |
p_n <- dnorm(x, N*p, sqrt(N*p*(1-p))); | |
c_b <- pbinom(x, N, p); | |
c_n <- pnorm(x, N*p, sqrt(N*p*(1-p))); | |
x2 <- linspace(5-2.5,5+2.5,1001); | |
q2 <- dnorm(x2,N*p, sqrt(N*p*(1-p))); | |
plot(x, p_n, col="red") | |
polygon(c(min(x2), x2, max(x2)), c(0, q2, 0), col='lightblue') | |
lines(x, p_b, col="black", type="h") | |
# How moment generating of Gaussian approximates in CLT | |
library(pracma) | |
p <- 0.5 | |
s <- seq(-10,10,length.out=1001) | |
MX <- 1-p+p*exp(s) | |
N <- 2 | |
semilogy(s, (1-p+p*exp(s/N))**N, lwd=4, col="lightblue", xlim=c(-10,10), ylim=c(10**-2, 10**5)) | |
mu <- p | |
sigma <- sqrt(p*(1-p)/N); | |
MZ <- exp(mu*s + sigma^2*s**2/2); | |
lines(s, MZ, lwd=5, lty=3); | |
legend("topleft", c('Binomial MGF', 'Gaussian MGF'), fill=c('lightblue', 'black')) | |
# Failure of Central Limit Theorem at tails | |
library(pracma) | |
x <- seq(-1,5,length.out=1001) | |
lambda <- 1 | |
N <- 1 | |
f1 <- (N**(1/2)/lambda)*dgamma((x+sqrt(N))/(lambda/sqrt(N)), N, lambda) | |
semilogy(x, f1, lwd=0.5, col='lightgray', xlim=c(-1,5), ylim=c(10**-6, 1)) | |
N <- 10 | |
f1 <- (N**(1/2)/lambda)*dgamma((x+sqrt(N))/(lambda/sqrt(N)), N, lambda) | |
lines(x, f1, lwd=4, col='gray') | |
N <- 100 | |
f1 <- (N**(1/2)/lambda)*dgamma((x+sqrt(N))/(lambda/sqrt(N)), N, lambda) | |
lines(x, f1, lwd=4, col='darkgray') | |
N <- 1000 | |
f1 <- (N**(1/2)/lambda)*dgamma((x+sqrt(N))/(lambda/sqrt(N)), N, lambda) | |
lines(x, f1, lwd=4, col='black') | |
g <- dnorm(x,0,1) | |
lines(x, g, lwd=4, lty=6, col='red') | |
legend("bottomleft", c('N=1', 'N=10', 'N=100', 'N=1000', 'Gaussian'), fill=c('lightgray', 'gray', 'darkgray', 'black', 'red')) | |