205 lines
4.8 KiB
XML
205 lines
4.8 KiB
XML
<?xml version="1.0" encoding="UTF-8"?>
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<!-- When creating a new activity, make a copy of this file with appropriate name -->
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<worksheet xml:id="quiz-02">
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<title>Quiz 2</title>
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<!-- Optional introduction -->
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<introduction>
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<p>
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The following work should be completed individually.
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Use of notes or textbooks is not allowed.
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You may use a scientific calculator, not a graphing calculator or phone app.
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</p>
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<p>
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Show all work unless instructed otherwise.
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</p>
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</introduction>
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<page>
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<!-- Exercises start here. -->
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<exercise>
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<introduction>
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<p>
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Write either True or False for each of the following statements.
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No justification is required.
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</p>
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</introduction>
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<task>
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<statement>
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<p>
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Let <m>n, k</m> be integers with <m>0 \leq k \leq n</m>.
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Then <m>{n\choose k} = {n \choose n - k}</m>.
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</p>
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</statement>
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<solution>
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<p>
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True.
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</p>
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</solution>
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</task>
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<task>
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<statement>
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<p>
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Suppose <m>X</m> is a continuous random variable with pdf <m>f(x)</m> and cdf <m>F(x)</m>.
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Then <m>\displaystyle{\int_a^b f(x)\ dx = F(b) - F(a)}</m>.
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</p>
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</statement>
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<solution>
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<p>
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True.
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</p>
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</solution>
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</task>
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<task>
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<statement>
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<p>
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Suppose <m>X</m> is a random variable taking values between 0 and 6.
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Then <m>\E(X) = 3</m>.
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</p>
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</statement>
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<solution>
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<p>
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False.
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</p>
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</solution>
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</task>
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</exercise>
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<exercise>
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<introduction>
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<p>
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Consider the joint distribution for <m>X</m> and <m>Y</m> below.
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</p>
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<table>
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<title>Joint Distribution</title>
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<tabular halign="center">
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<row bottom="minor">
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<cell right="minor"></cell>
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<cell><m>X = 0</m></cell>
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<cell><m>X = 1</m></cell>
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<cell><m>X = 2</m></cell>
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</row>
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<row>
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<cell right="minor"><m>Y = 0</m></cell>
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<cell>0.1</cell>
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<cell>0.05</cell>
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<cell>0.1</cell>
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</row>
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<row>
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<cell right="minor"><m>Y = 1</m></cell>
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<cell>0.3</cell>
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<cell>0.15</cell>
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<cell>0.3</cell>
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</row>
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</tabular>
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</table>
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</introduction>
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<task>
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<statement>
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<p>
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Find the marginal distributions for <m>X</m> and <m>Y</m>.
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</p>
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</statement>
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<solution>
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<p>
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<md>
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<mrow> \Pr(X = 0) \amp = 0.1 + 0.3 = 0.4 </mrow>
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<mrow> \Pr(X = 1) \amp = 0.05 + 0.15 = 0.2 </mrow>
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<mrow> \Pr(X = 2) \amp = 0.1 + 0.3 = 0.4 </mrow>
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<mrow> \Pr(Y = 0) \amp = 0.1 + 0.05 + 0.1 = 0.25 </mrow>
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<mrow> \Pr(Y = 1) \amp = 0.3 + 0.15 + 0.3 = 0.75 </mrow>
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</md>
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</p>
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</solution>
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</task>
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<task>
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<statement>
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<p>
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Are <m>X</m> and <m>Y</m> independent?
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</p>
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</statement>
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<solution>
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<p>
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<md>
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<mrow> \Pr(X = 2, Y = 0) \amp = 0.2 </mrow>
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<mrow> \Pr(X = 2)\Pr(Y = 0) \amp = (0.4)(0.25) = 0.1 </mrow>
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</md>
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Since <m>0.2 \neq 0.1</m>, <m>X</m> and <m>Y</m> are not independent.
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</p>
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</solution>
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</task>
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</exercise>
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</page>
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<page>
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<exercise>
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<introduction>
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<p>
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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>.
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</p>
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</introduction>
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<task>
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<statement>
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<p>
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Find the pdf <m>f(x)</m>.
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</p>
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</statement>
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<solution>
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<p>
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<md>
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<mrow> f(x) = F'(x) = 4x - 4x^3. </mrow>
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</md>
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</p>
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</solution>
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</task>
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<task>
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<statement>
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<p>
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Find <m>\E(X)</m>.
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</p>
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</statement>
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<solution>
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<p>
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<md>
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<mrow> \E(X) \amp = \int_0^1 x f(x)\ dx </mrow>
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<mrow> \amp = \int_0^1 x (4x - 4x^3)\ dx </mrow>
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<mrow> \amp = \int_0^1 4x^2 - 4x^4\ dx </mrow>
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<mrow> \amp = \frac{4x^2}{3} - \frac{4x^5}{5}\bigg|_0^1 </mrow>
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<mrow> \amp = \left(\frac{4}{3} - \frac{4}{5}\right) - (0) </mrow>
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<mrow> \amp = \frac{8}{15} \approx 0.533. </mrow>
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</md>
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</p>
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</solution>
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</task>
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</exercise>
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</page>
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</worksheet> |