Update to PreTeXt project source.

source/review/Exam-1-Review.ptx

@@ -356,17 +356,17 @@
       </p>
     </answer>
   </exercise>
-
-  <!-- <exercise>
+  <!--
+  <exercise>
     <statement>
       <p>
         Each of 10 toxin molecules inside a cell has a 0.3 probability of leaving the cell during a 1-minute period.
         For each value of <m>n = 1, 2, 3, \dotsc</m>, and for each value of <m>0\leq k \leq n</m>, find the probability that exactly <m>k</m> toxin molecules remain in the cell after the <m>n</m>th minute.
       </p>
     </statement>
-  </exercise> -->
+  </exercise>
 
-  <!-- TODO write a solution --> <exercisegroup>
+  --> <!-- TODO write a solution --> <exercisegroup>
   <introduction>
     <p>
       In each of the following scenarios with given events <m>A</m> and <m>B</m>, alculate <m>\Pr(A), \Pr(B)</m>, <m>\Pr(A\cap B)</m>, <m>\Pr(A \mid B)</m>, and <m>\Pr(B \mid A)</m>.
@@ -1126,5 +1126,45 @@
       </p>
     </statement>
   </exercise>
+
+  <exercise>
+    <statement>
+      <p>
+        Suppose a coin has an unknown probability of coming up heads.
+        We perform the experiment in <m>n</m> independent trials, during which it takes <m>k_1, k_2, \dotsc, k_n</m> flips to see our first heads in each trial.
+        Find a "common sense" MLE formula for the geometric distribution.
+      </p>
+    </statement>
+  </exercise>
+
+  <exercise>
+    <statement>
+      <p>
+        A particular store owner wants to approximate the average hourly rate at which customers come into the store.
+        They observe 80 customers enter during a particular 4-hour shift.
+        What is the maximum likelihood estimation for the hourly customer rate?
+      </p>
+    </statement>
+  </exercise>
+
+  <exercise>
+    <statement>
+      <p>
+        A radioactive material emits particles at an unknown probabilistic rate <m>\lambda</m> particles per minute.
+        We observe particles emitted at times 1.1, 1.7, 1.3, 2.2, 1.9, and 1.8 minutes.
+        Write the likelihood function <m>\mathcal{L}(\lambda)</m> based on this data.
+        What is the maximum likelihood estimation for <m>\lambda</m>?
+      </p>
+    </statement>
+  </exercise>
+
+  <exercise>
+    <statement>
+      <p>
+        Suppose a parameter <m>\theta</m> takes values in <m>[0, 1]</m> with likelihood function <m>\mathcal{L}(\theta) = \sqrt{\theta} - \theta^2</m>.
+        Find the maximum likelihood estimation of <m>\theta</m>.
+      </p>
+    </statement>
+  </exercise>
   </exercises>
 </section>