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Quiz 2 solutions

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<?xml version="1.0" encoding="UTF-8"?>
<!-- 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 </mrow>
<mrow> \Pr(X = 1) \amp = 0.05 + 0.15 = 0.2 </mrow>
<mrow> \Pr(X = 2) \amp = 0.1 + 0.3 = 0.4 </mrow>
<mrow> \Pr(Y = 0) \amp = 0.1 + 0.05 + 0.1 = 0.25 </mrow>
<mrow> \Pr(Y = 1) \amp = 0.3 + 0.15 + 0.3 = 0.75 </mrow>
</md>
</p>
</solution>
</task>
<task>
<statement>
<p>
Are <m>X</m> and <m>Y</m> independent?
</p>
</statement>
<solution>
<p>
<md>
<mrow> \Pr(X = 2, Y = 0) \amp = 0.2 </mrow>
<mrow> \Pr(X = 2)\Pr(Y = 0) \amp = (0.4)(0.25) = 0.1 </mrow>
</md>
Since <m>0.2 \neq 0.1</m>, <m>X</m> and <m>Y</m> are not 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>