From 18c992aa9d1592efb5dc476db7e29fdb08e1d158 Mon Sep 17 00:00:00 2001
From: Ramon Araujo <ramoncfaraujo@gmail.com>
Date: Mon, 13 Oct 2025 22:27:07 -0300
Subject: [PATCH] add solution to problem 5.15

---
 .../uniform_and_universality/index.tex        |  2 ++
 .../uniform_and_universality/problems/15.tex  | 20 +++++++++++++++++++
 2 files changed, 22 insertions(+)
 create mode 100644 src/chapters/5/sections/uniform_and_universality/problems/15.tex

diff --git a/src/chapters/5/sections/uniform_and_universality/index.tex b/src/chapters/5/sections/uniform_and_universality/index.tex
index 8e94ee3f..eff51dca 100644
--- a/src/chapters/5/sections/uniform_and_universality/index.tex
+++ b/src/chapters/5/sections/uniform_and_universality/index.tex
@@ -2,3 +2,5 @@ \section{Uniform and Universality}
 
 \subsection{problem 13}
 \input{problems/13}
+\subsection{problem 15}
+\input{problems/15}
diff --git a/src/chapters/5/sections/uniform_and_universality/problems/15.tex b/src/chapters/5/sections/uniform_and_universality/problems/15.tex
new file mode 100644
index 00000000..a61dba4f
--- /dev/null
+++ b/src/chapters/5/sections/uniform_and_universality/problems/15.tex
@@ -0,0 +1,20 @@
+Let $F$ be the CDF of $X$.
+It is given by $F(x) = 1 - e^{-\lambda x}$, for $x>0$.
+
+As per the Universality of the Uniform (UoU), $X$ can be generated from $U$ using $X$'s quantile function:
+
+$$
+X = F^{-1}(U)
+$$
+
+The quantile function is calculated as
+
+$$
+F^{-1}(x) = -\frac{1}{\lambda} \log(1-x)
+$$
+
+Plugging the quantile function into the UoU formula
+
+$$
+X = -\frac{1}{\lambda} \log(1 - U)
+$$