From 57175dbf3472fc3a12b23b39010f779c40cb40b2 Mon Sep 17 00:00:00 2001
From: aaaakshat <33050725+aaaakshat@users.noreply.github.com>
Date: Thu, 21 Oct 2021 12:06:24 -0400
Subject: [PATCH 1/7] Add pdf/cdf of a urv

---
 ch04.r | 12 ++++++++++++
 1 file changed, 12 insertions(+)
 create mode 100644 ch04.r

diff --git a/ch04.r b/ch04.r
new file mode 100644
index 0000000..c0ee293
--- /dev/null
+++ b/ch04.r
@@ -0,0 +1,12 @@
+# Chapter 4.3 (Cumulative Distribution Function)
+
+## CDF of a Gaussian mixture
+
+# R code to generate the PDF and CDF
+x = seq(-5, 10, (15)/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)

From 73aff20a439289f3b2472a5c80fcb053eb6758b8 Mon Sep 17 00:00:00 2001
From: aaaakshat <33050725+aaaakshat@users.noreply.github.com>
Date: Thu, 21 Oct 2021 12:28:09 -0400
Subject: [PATCH 2/7] Add pdf/cdf of ERV

---
 ch04.r | 15 +++++++++++++--
 1 file changed, 13 insertions(+), 2 deletions(-)

diff --git a/ch04.r b/ch04.r
index c0ee293..138d3bd 100644
--- a/ch04.r
+++ b/ch04.r
@@ -1,12 +1,23 @@
 # Chapter 4.3 (Cumulative Distribution Function)
 
-## CDF of a Gaussian mixture
+## 4.10 CDF of a Uniform Random Variable
 
 # R code to generate the PDF and CDF
-x = seq(-5, 10, (15)/1500)
+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)

From 8afad0932064f19b2ecbc7f97b507c800c67fe98 Mon Sep 17 00:00:00 2001
From: aaaakshat <33050725+aaaakshat@users.noreply.github.com>
Date: Thu, 21 Oct 2021 15:42:06 -0400
Subject: [PATCH 3/7] Add URM stat functions

---
 ch04.r | 36 ++++++++++++++++++++++++++++++++++++
 1 file changed, 36 insertions(+)

diff --git a/ch04.r b/ch04.r
index 138d3bd..fd93c98 100644
--- a/ch04.r
+++ b/ch04.r
@@ -21,3 +21,39 @@ 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)
+

From d85ba280ba0515ee030973100686741e79ccb2f5 Mon Sep 17 00:00:00 2001
From: aaaakshat <33050725+aaaakshat@users.noreply.github.com>
Date: Thu, 21 Oct 2021 16:43:59 -0400
Subject: [PATCH 4/7] Add PDF and CDF of exponential RV

---
 ch04.r | 26 ++++++++++++++++++++++++++
 1 file changed, 26 insertions(+)

diff --git a/ch04.r b/ch04.r
index fd93c98..98c4c9d 100644
--- a/ch04.r
+++ b/ch04.r
@@ -57,3 +57,29 @@ V = unifstat(a, b)$var
 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)
+
+

From bd54518db54e2c091dbe1026e5cd4db2ccbd02a9 Mon Sep 17 00:00:00 2001
From: aaaakshat <33050725+aaaakshat@users.noreply.github.com>
Date: Thu, 21 Oct 2021 17:21:13 -0400
Subject: [PATCH 5/7] Add chapter 4.6

---
 ch04.r | 37 +++++++++++++++++++++++++++++++++++++
 1 file changed, 37 insertions(+)

diff --git a/ch04.r b/ch04.r
index 98c4c9d..de58f57 100644
--- a/ch04.r
+++ b/ch04.r
@@ -82,4 +82,41 @@ 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(2.5,17.5))
 

From ced2ac59676a642e43dd6647e93f951b73b3cbac Mon Sep 17 00:00:00 2001
From: aaaakshat <33050725+aaaakshat@users.noreply.github.com>
Date: Thu, 21 Oct 2021 21:39:59 -0400
Subject: [PATCH 6/7] Fix hist(Z) for gaussian RV

---
 ch04.r | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/ch04.r b/ch04.r
index de58f57..6340d7a 100644
--- a/ch04.r
+++ b/ch04.r
@@ -118,5 +118,5 @@ X1 = runif(N, 1, 6)
 X2 = runif(N, 1, 6)
 X3 = runif(N, 1, 6)
 Z = X1 + X2 + X3
-hist(Z, breaks=seq(2.5,17.5))
+hist(Z, breaks=seq(2.5,18.5))
 

From 800e4016e40259240575e48b442da01491cced10 Mon Sep 17 00:00:00 2001
From: aaaakshat <33050725+aaaakshat@users.noreply.github.com>
Date: Thu, 21 Oct 2021 22:13:10 -0400
Subject: [PATCH 7/7] Add section 4.8

---
 ch04.r | 26 ++++++++++++++++++++++++++
 1 file changed, 26 insertions(+)

diff --git a/ch04.r b/ch04.r
index 6340d7a..22a4fc6 100644
--- a/ch04.r
+++ b/ch04.r
@@ -120,3 +120,29 @@ 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)