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Setting up Exam 1 Review

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andy
2026-02-10 08:19:40 -05:00
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source/quizzes/quiz-02.ptx
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<solution>
<p>
<m>\Pr(X = 2, Y = 0) = 0.2 </m>, and <m> \Pr(X = 2)\Pr(Y = 0) = (0.4)(0.25) = 0.1</m>. Since <m>0.2 \neq 0.1</m>, <m>X</m> and <m>Y</m> are not independent.
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>