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Adding some answers

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andy
2026-01-29 06:35:50 -05:00
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source/sec-Joint-Distributions.ptx
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@@ -120,9 +120,84 @@
<exercise>
<statement>
<p>
TODO
Consider <m>X, Y</m> with the joint distribution table below.
Are <m>X, Y</m> independent?
</p>
<table>
<title>Joint distribution for <m>X, Y</m></title>
<tabular halign="center">
<row bottom="minor">
<cell right="minor"></cell>
<cell right="minor"><m>X = 0</m></cell>
<cell><m>X = 1</m></cell>
</row>
<row bottom="minor">
<cell right="minor"><m>Y = 0</m></cell>
<cell right="minor">0.2</cell>
<cell>0.3</cell>
</row>
<row>
<cell right="minor"><m>Y = 1</m></cell>
<cell right="minor">0.4</cell>
<cell>0.1</cell>
</row>
</tabular>
</table>
</statement>
<answer>
<p>
No.
For example, <m>\Pr(X = 0, Y = 0) \neq \Pr(X = 0)\Pr(Y = 0).</m>
</p>
</answer>
</exercise>
<exercise>
<statement>
<p>
Suppose <m>X, Y</m> have the distributions:
<md>
<mrow> \Pr(X = 0) \amp = 0.1 \amp \Pr(Y = 0) \amp = 0.4 </mrow>
<mrow> \Pr(X = 1) \amp = 0.4 \amp \Pr(Y = 1) \amp = 0.6 </mrow>
<mrow> \Pr(X = 2) \amp = 0.5 </mrow>
</md>
Assuming <m>X, Y</m> are independent, write a joint distribution table.
</p>
</statement>
<answer>
<table>
<title>Joint Distribution</title>
<tabular halign="center">
<row bottom="minor">
<cell right="minor"></cell>
<cell right="minor"><m>X = 0</m></cell>
<cell right="minor"><m>X = 1</m></cell>
<cell><m>X = 2</m></cell>
</row>
<row bottom="minor">
<cell right="minor"><m>Y = 0</m></cell>
<cell right="minor">0.04</cell>
<cell right="minor">0.16</cell>
<cell>0.2</cell>
</row>
<row>
<cell right="minor"><m>Y = 1</m></cell>
<cell right="minor">0.06</cell>
<cell right="minor">0.24</cell>
<cell>0.3</cell>
</row>
</tabular>
</table>
</answer>
</exercise>
</exercises>
</section>