Merge pull request #1 from bishta/chap-04

Add Chapter 4

ch04.r

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+# Chapter 4.3 (Cumulative Distribution Function)
+
+## 4.10 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)
+plot(x, F, type="n")
+lines(x, F, lwd=5)
+
+## 4.11 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)
+plot(x, F, type="n")
+lines(x, F, lwd=5)
+
+# Chapter 4.5
+
+# Generate a uniform random variable
+
+# R code to generate 1000 uniform random numbers
+a = 0; b = 1;
+X = runif(1000, a, b)
+hist(X)
+
+# Mean, variance, median, mode of a uniform random variable
+
+# R code to computer empirical mean, var, median, mode
+library(pracma)
+a = 0; b = 1;
+X = runif(1000, a, b)
+M = mean(X)
+V = var(X)
+Med = median(X)
+Mod = Mode(X)
+
+# R code to compute mean and variance
+unifstat = function(a, b) {
+    M = (a+b)/2
+    V = ((a-b)^2)/12
+    return(list(mean = M, var = V))
+}
+
+a = 0; b = 1;
+M = unifstat(a, b)$mean
+V = unifstat(a, b)$var
+
+# 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
+
+# R code to plot the exponential PDF
+lambda1 = 1/2
+lambda2 = 1/5
+x = seq(0, 1, (0+1)/1000)
+f1 = dexp(x, 1/lambda1)
+f2 = dexp(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)
+
+# 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)
+
+# Chapter 4.6
+
+## PDF and CDF of a Gaussian random variable
+
+# 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")
+
+# 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)
+
+# 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
+
+# R code to compute skewness and kurtosis
+library(e1071)
+X = rgamma(10000, 3, 5)
+s = skewness(X)
+k = kurtosis(X)
+
+# 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))
+
+# Chapter 4.8
+
+## Generating Gaussians from uniform
+
+# R code to generate Gaussian from uniform
+library(HDInterval)
+mu = 3
+sigma = 2
+U = runif(10000, 0, 1)
+gU = sigma * inverseCDF(U, pnorm) + mu;
+hist(U)
+hist(gU)
+
+# R code to generate exponential random variables
+lambda = 1
+U = runif(10000, 0, 1)
+gU = -(1/lambda)*log(1-U)
+
+# R code to generate the desired random variables
+U = runif(10000, 0, 1)
+gU = rep(0, 10000)
+gU[U >= 0.0 & U <= 0.1] = 1
+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)