From e6cea0acef99d94268b8395cba66a9dd225647df Mon Sep 17 00:00:00 2001
From: andyeisenberg <andyjeisenberg@gmail.com>
Date: Thu, 17 Sep 2026 13:18:23 -0400
Subject: [PATCH] Variance exercises

---
 source/sec-Continuous-RVs.ptx |  2 +-
 source/sec-Expected-Value.ptx |  6 +++---
 source/sec-Variance.ptx       | 38 ++++++++++++++++++++++++++++++++++-
 3 files changed, 41 insertions(+), 5 deletions(-)

diff --git a/source/sec-Continuous-RVs.ptx b/source/sec-Continuous-RVs.ptx
index d3a250a..0e38ea2 100644
--- a/source/sec-Continuous-RVs.ptx
+++ b/source/sec-Continuous-RVs.ptx
@@ -302,7 +302,7 @@
   </subsection>
 
   <exercises xml:id="exercises-Continuous-RVs">
-  <exercise>
+  <exercise xml:id="exercise-continuous-RV-find-k">
     <introduction>
       <p>
         A continuous random variable <m>X</m> taking values in <m>[1, 4]</m> has p.d.f.
diff --git a/source/sec-Expected-Value.ptx b/source/sec-Expected-Value.ptx
index 0c842e0..ef2e7d3 100644
--- a/source/sec-Expected-Value.ptx
+++ b/source/sec-Expected-Value.ptx
@@ -624,14 +624,14 @@
       <p>
         A continuous random variable <m>X</m> taking values in <m>[1, 4]</m> has p.d.f.
         <m>f(x) = k(x - \sqrt{x})</m> for some constant <m>k</m>.
-        In a previous problem, you found the value of <m>k</m>.
+        In a previous problem (<xref ref="exercise-continuous-RV-find-k"/>), you found the value of <m>k</m>.
         Now, find <m>\E(X)</m>.
       </p>
     </statement>
 
     <answer>
       <p>
-        <m>E(X) \approx 3.04.</m>
+        <m>\frac{258}{85} \approx 3.04.</m>
       </p>
     </answer>
   </exercise>
@@ -647,7 +647,7 @@
 
     <answer>
       <p>
-        <m>\frac{1}{2}\left( \ln(2) + \frac{7}{3}\right) \approx 1.513.</m>
+        <m>\frac{1}{2}\left( \ln(2) + \frac{7}{3}\right) \approx 1.513</m>.
       </p>
     </answer>
   </exercise>
diff --git a/source/sec-Variance.ptx b/source/sec-Variance.ptx
index 5f37827..91a98dc 100644
--- a/source/sec-Variance.ptx
+++ b/source/sec-Variance.ptx
@@ -248,6 +248,12 @@
           </tabular>
         </table>
       </statement>
+
+      <answer>
+        <p>
+          <m>5.76</m>.
+        </p>
+      </answer>
     </exercise>
 
     <exercise>
@@ -299,6 +305,12 @@
           What is <m>\Var(X)</m>?
         </p>
       </statement>
+
+      <answer>
+        <p>
+          <m>1/18</m>.
+        </p>
+      </answer>
     </exercise>
 
     <exercise>
@@ -309,6 +321,12 @@
           What is <m>\Var(X)</m>?
         </p>
       </statement>
+
+      <answer>
+        <p>
+          <m>0.6</m>.
+        </p>
+      </answer>
     </exercise>
 
     <exercise>
@@ -319,6 +337,12 @@
           What is <m>\Var(X)</m>?
         </p>
       </statement>
+
+      <answer>
+        <p>
+          <m>4 - \left(\frac{4}{3}\ln(4)\right)^2 \approx 0.583</m>.
+        </p>
+      </answer>
     </exercise>
 
     <exercise>
@@ -326,10 +350,16 @@
         <p>
           A continuous random variable <m>X</m> taking values in <m>[1, 4]</m> has p.d.f.
           <m>f(x) = k(x - \sqrt{x})</m> for some constant <m>k</m>.
-          In a previous problem, you found the value of <m>k</m>.
+          In a previous problem (<xref ref="exercise-continuous-RV-find-k"/>), you found the value of <m>k</m>.
           Now, find <m>\Var(X)</m>.
         </p>
       </statement>
+
+      <answer>
+        <p>
+          <m>\frac{2307}{238} - \left(\frac{258}{85}\right)^2 \approx 0.48</m>.
+        </p>
+      </answer>
     </exercise>
 
     <exercise>
@@ -340,6 +370,12 @@
           Find <m>\Var(X)</m>.
         </p>
       </statement>
+
+      <answer>
+        <p>
+          <m>\frac{29}{24} - \frac{1}{2}\ln(2) \approx 0.862</m>.
+        </p>
+      </answer>
     </exercise>
     </exercises>
   </section>
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