Fall-2026-Math-1041/source/quizzes/quiz-02.ptx

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<!-- When creating a new activity, make a copy of this file with appropriate name -->
<worksheet xml:id="quiz-02">
  <title>Quiz 2</title>

  <!-- Optional introduction -->
  <introduction>
    <p>
      The following work should be completed individually.
      Use of notes or textbooks is not allowed.
      You may use a scientific calculator, not a graphing calculator or phone app.
    </p>

    <p>
      Show all work unless instructed otherwise.
    </p>
  </introduction>


  <page>
    <!-- Exercises start here. -->
    <exercise>
      <introduction>
        <p>
          Write either True or False for each of the following statements.
          No justification is required.
        </p>
      </introduction>


      <task>
        <statement>
          <p>
            Let <m>n, k</m> be integers with <m>0 \leq k \leq n</m>.
            Then <m>{n\choose k} = {n \choose n - k}</m>.
          </p>
        </statement>

        <solution>
          <p>
            True.
          </p>
        </solution>
      </task>


      <task>
        <statement>
          <p>
            Suppose <m>X</m> is a continuous random variable with pdf <m>f(x)</m> and cdf <m>F(x)</m>.
            Then <m>\displaystyle{\int_a^b f(x)\ dx = F(b) - F(a)}</m>.
          </p>
        </statement>

        <solution>
          <p>
            True.
          </p>
        </solution>
      </task>


      <task>
        <statement>
          <p>
            Suppose <m>X</m> is a random variable taking values between 0 and 6.
            Then <m>\E(X) = 3</m>.
          </p>
        </statement>

        <solution>
          <p>
            False.
          </p>
        </solution>
      </task>
    </exercise>

    <exercise>
      <introduction>
        <p>
          Consider the joint distribution for <m>X</m> and <m>Y</m> below.
        </p>

        <table>
          <title>Joint Distribution</title>

          <tabular halign="center">
            <row bottom="minor">
              <cell right="minor"></cell>
              <cell><m>X = 0</m></cell>
              <cell><m>X = 1</m></cell>
              <cell><m>X = 2</m></cell>
            </row>

            <row>
              <cell right="minor"><m>Y = 0</m></cell>
              <cell>0.1</cell>
              <cell>0.05</cell>
              <cell>0.1</cell>
            </row>

            <row>
              <cell right="minor"><m>Y = 1</m></cell>
              <cell>0.3</cell>
              <cell>0.15</cell>
              <cell>0.3</cell>
            </row>
          </tabular>
        </table>
      </introduction>


      <task>
        <statement>
          <p>
            Find the marginal distributions for <m>X</m> and <m>Y</m>.
          </p>
        </statement>

        <solution>
          <p>
            <md>
              <mrow> \Pr(X = 0) \amp = 0.1 + 0.3 = 0.4 \amp \Pr(Y = 0) \amp = 0.1 + 0.05 + 0.1 = 0.25 </mrow>
              <mrow> \Pr(X = 1) \amp = 0.05 + 0.15 = 0.2 \amp \Pr(Y = 1) \amp = 0.3 + 0.15 + 0.3 = 0.75 </mrow>
              <mrow> \Pr(X = 2) \amp = 0.1 + 0.3 = 0.4 </mrow>
            </md>
          </p>
        </solution>
      </task>


      <task>
        <statement>
          <p>
            Are <m>X</m> and <m>Y</m> independent?
          </p>
        </statement>

        <solution>
          <p>
            Checking each cell in the table:
            <md>
              <mrow> \Pr(X = 0, Y = 0) \amp = 0.1 = (0.4)(0.25) = \Pr(X = 0)\Pr(Y = 0) </mrow>
              <mrow> \Pr(X = 1, Y = 0) \amp = 0.05 = (0.2)(0.25) = \Pr(X = 1)\Pr(Y = 0) </mrow>
              <mrow> \Pr(X = 2, Y = 0) \amp = 0.1 = (0.4)(0.25) = \Pr(X = 2)\Pr(Y = 0) </mrow>
              <mrow> \Pr(X = 0, Y = 1) \amp = 0.3 = (0.4)(0.75) = \Pr(X = 0)\Pr(Y = 1) </mrow>
              <mrow> \Pr(X = 1, Y = 1) \amp = 0.15 = (0.2)(0.75) = \Pr(X = 1)\Pr(Y = 1) </mrow>
              <mrow> \Pr(X = 2, Y = 1) \amp = 0.3 = (0.4)(0.75) = \Pr(X = 2)\Pr(Y = 1) </mrow>
            </md>
            So <m>X, Y</m> are independent.
          </p>
        </solution>
      </task>
    </exercise>
  </page>


  <page>
    <exercise>
      <introduction>
        <p>
          Suppose a continuous random variable <m>X</m> taking values in <m>[0, 1]</m> has cdf <m>F(x) = 2x^2 - x^4</m>.
        </p>
      </introduction>


      <task>
        <statement>
          <p>
            Find the pdf <m>f(x)</m>.
          </p>
        </statement>

        <solution>
          <p>
            <md>
              <mrow> f(x) = F'(x) = 4x - 4x^3. </mrow>
            </md>
          </p>
        </solution>
      </task>


      <task>
        <statement>
          <p>
            Find <m>\E(X)</m>.
          </p>
        </statement>

        <solution>
          <p>
            <md>
              <mrow> \E(X) \amp = \int_0^1 x f(x)\ dx </mrow>
              <mrow> \amp = \int_0^1 x (4x - 4x^3)\ dx </mrow>
              <mrow> \amp = \int_0^1 4x^2 - 4x^4\ dx </mrow>
              <mrow> \amp =  \frac{4x^2}{3} - \frac{4x^5}{5}\bigg|_0^1 </mrow>
              <mrow> \amp =  \left(\frac{4}{3} - \frac{4}{5}\right) - (0) </mrow>
              <mrow> \amp =  \frac{8}{15} \approx 0.533. </mrow>
            </md>
          </p>
        </solution>
      </task>
    </exercise>
  </page>
</worksheet>