Reformat Chapter 4

ch04.r

@@ -1,37 +1,61 @@
+#############
 # Chapter 4.3 (Cumulative Distribution Function)
 
-## 4.10 CDF of a Uniform Random Variable
+## CDF of a Gaussian Mixture
+
+# R code to generate the PDF and CDF
+library(EnvStats)
+
+x = seq(-5, 10, (5+10)/1000)
+x = sort(x)
+f = dnormMix(x, 0, 1, 5, 1, p.mix=c(0.7))
+plot(x,f, xlim=c(-5,10), ylim=c(0,0.4), type="n")
+lines(x, f, lwd=4, col="blue")
+polygon(c(min(x), x[x<=0.8]), c(f[x<=0.8], 0), col="lightblue")
+grid()
+
+F = pnormMix(x, 0, 1, 5, 1, p.mix=c(0.7))
+plot(x, F, xlim=c(-5,10), ylim=c(0,0.4), type="n")
+grid()
+
+## CDF of a Uniform random variable
 
 # R code to generate the PDF and CDF
 x = seq(-5, 10, (5+10)/1500)
 f = dunif(x, -3, 4)
 F = punif(x, -3, 4)
 plot(x, f, type="n")
 lines(x, f, lwd=5)
+grid()
 plot(x, F, type="n")
 lines(x, F, lwd=5)
+grid()
 
-## 4.11 CDF of an exponential random variable
+## CDF of an exponential random variable
 
 # R code to generate the PDF and CDF
 x = seq(-5, 10, (5+10)/1500)
 f = dexp(x, 1/2)
 F = pexp(x, 1/2)
 plot(x, f, type="n")
 lines(x, f, lwd=5)
+grid()
 plot(x, F, type="n")
 lines(x, F, lwd=5)
+grid()
 
+#############
 # Chapter 4.5
 
-# Generate a uniform random variable
+## Generate a uniform random variable
 
 # R code to generate 1000 uniform random numbers
 a = 0; b = 1;
 X = runif(1000, a, b)
 hist(X)
+grid()
 
-# Mean, variance, median, mode of a uniform random variable
+## Mean, variance, median, mode of a uniform random variable
 
 # R code to computer empirical mean, var, median, mode
 library(pracma)
@@ -42,6 +66,8 @@ V = var(X)
 Med = median(X)
 Mod = Mode(X)
 
+## Mean and Variance computation (not on "Code and Data" section of website, but written in textbook p.34)
+
 # R code to compute mean and variance
 unifstat = function(a, b) {
     M = (a+b)/2
@@ -53,14 +79,16 @@ a = 0; b = 1;
 M = unifstat(a, b)$mean
 V = unifstat(a, b)$var
 
+## Probability of a uniform random variable
+
 # R code to compute the probability P(0.2 < X < 0.3)
 a = 0; b = 1;
 F = punif(0.3, a, b) - punif(0.2, a, b)
 
 
-## PDF and CDF of an exponential random variable
+## PDF of an exponential random variable
 
-# R code to plot the exponential PDF
+# R code to generate the PDF and CDF of an exponential random variable
 lambda1 = 1/2
 lambda2 = 1/5
 x = seq(0, 1, (0+1)/1000)
@@ -70,56 +98,82 @@ plot(x, f2, type="n")
 lines(x, f1, lwd=4, col="blue")
 lines(x, f2, lwd=4, col="red")
 legend(0, 1, legend=c(expression(paste(lambda, "=5")), expression(paste(lambda, "=2"))), col=c("red", "blue"), lty=1:1)
+grid()
 
-# R code to plot the exponential CDF
-lambda1 = 1/2
-lambda2 = 1/5
-x = seq(0, 1, (0+1)/1000)
 F1 = pexp(x, 1/lambda1)
 F2 = pexp(x, 1/lambda2)
 plot(x, F2, type="n")
 lines(x, F1, lwd=4, col="blue")
 lines(x, F2, lwd=4, col="red")
 legend(0, 1, legend=c(expression(paste(lambda, "=5")), expression(paste(lambda, "=2"))), col=c("red", "blue"), lty=1:1)
+grid()
 
+#############
 # Chapter 4.6
 
-## PDF and CDF of a Gaussian random variable
+## Generate Gaussian PDF (not on "Code and Data" section on website, but written in textbook p.42)
 
 # R code to generate a Gaussian PDF
 x = seq(-10, 10, (10+10)/1000)
 mu = 0; sigma = 1;
 f = dnorm(x, mu, sigma)
 plot(x, f, type="n")
 lines(x, f, lwd=4, col="blue")
+grid()
+
+## PDF and CDF of a Gaussian random variable
 
 # R code to generate standard Gaussian PDF and CDF
 x = seq(-5, 5, (5+5)/1000)
 f = dnorm(x)
 F = pnorm(x)
-plot(x, f)
-plot(x, F)
+plot(x, f, type = "n")
+lines(x, f, lwd=6)
+grid()
+plot(x, F, type = "n")
+lines(x, F, lwd=6)
+grid()
+
+## Verify standardised gaussian (not on "Code and Data" section on website, but written in textbook p.45)
 
 # R code to verify standardised Gaussian
 x = seq(-5, 5, (5+5)/1000)
 mu = 3; sigma = 2;
 f1 = dnorm((x-mu)/sigma, 0, 1) # Standardised
 f2 = dnorm(x, mu, sigma) # Raw
 
+# Skewness and kurtosis of a random variable
+
+# R code to plot a Gamma distribution
+x = seq(0, 30, (0+30)/1000)
+theta = 1
+plot(x, dgamma(x, 2, theta), type = "n")
+lines(x, dgamma(x, 2, theta), lwd = 4)
+lines(x, dgamma(x, 5, theta), lwd = 4, col = "#333333")
+lines(x, dgamma(x, 10, theta), lwd = 4, col = "#666666")
+lines(x, dgamma(x, 15, theta), lwd = 4, col = "#999999")
+lines(x, dgamma(x, 20, theta), lwd = 4, col = "#CCCCCC")
+legend(23, 0.36, legend=c("k = 2", "k = 5", "k = 10", "k = 15", "k = 20"), col=c("1", "#333333", "#666666", "#999999", "#CCCCCC"), lwd = 4,lty=1:1)
+grid()
+
 # R code to compute skewness and kurtosis
 library(e1071)
 X = rgamma(10000, 3, 5)
 s = skewness(X)
 k = kurtosis(X)
 
+# Histogram of Z = X1 + X2 + X3 (not on "Code and "Data" section, but written in textbook p.50)
+
 # R code to show the histogram of Z = X1 + X2 + X3
 N = 10000
 X1 = runif(N, 1, 6)
 X2 = runif(N, 1, 6)
 X3 = runif(N, 1, 6)
 Z = X1 + X2 + X3
 hist(Z, breaks=seq(2.5,18.5))
+grid()
 
+#############
 # Chapter 4.8
 
 ## Generating Gaussians from uniform
@@ -131,13 +185,19 @@ sigma = 2
 U = runif(10000, 0, 1)
 gU = sigma * inverseCDF(U, pnorm) + mu;
 hist(U)
+grid()
 hist(gU)
+grid()
+
+## Generating exponential random variables from uniform random nums (not in "code" section, but in textbook p.63)
 
 # R code to generate exponential random variables
 lambda = 1
 U = runif(10000, 0, 1)
 gU = -(1/lambda)*log(1-U)
 
+## Plotting PDFs based on transformed vars (not in "code" section, but in textbook p.65)
+
 # R code to generate the desired random variables
 U = runif(10000, 0, 1)
 gU = rep(0, 10000)
@@ -146,3 +206,6 @@ gU[U > 0.1 & U <= 0.6] = 2
 gU[U > 0.6 & U <= 0.9] = 3
 gU[U > 0.9 & U <= 1] = 4
 hist(gU)
+grid()
+
+#############