diff --git a/source/review/Exam-1-Review.ptx b/source/review/Exam-1-Review.ptx index 8bc9e26..3aee09e 100644 --- a/source/review/Exam-1-Review.ptx +++ b/source/review/Exam-1-Review.ptx @@ -356,17 +356,17 @@

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In each of the following scenarios with given events A and B, alculate \Pr(A), \Pr(B), \Pr(A\cap B), \Pr(A \mid B), and \Pr(B \mid A). @@ -1126,5 +1126,45 @@

+ + + +

+ Suppose a coin has an unknown probability of coming up heads. + We perform the experiment in n independent trials, during which it takes k_1, k_2, \dotsc, k_n flips to see our first heads in each trial. + Find a "common sense" MLE formula for the geometric distribution. +

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+ + + +

+ A particular store owner wants to approximate the average hourly rate at which customers come into the store. + They observe 80 customers enter during a particular 4-hour shift. + What is the maximum likelihood estimation for the hourly customer rate? +

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+ + + +

+ A radioactive material emits particles at an unknown probabilistic rate \lambda particles per minute. + We observe particles emitted at times 1.1, 1.7, 1.3, 2.2, 1.9, and 1.8 minutes. + Write the likelihood function \mathcal{L}(\lambda) based on this data. + What is the maximum likelihood estimation for \lambda? +

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+ + + +

+ Suppose a parameter \theta takes values in [0, 1] with likelihood function \mathcal{L}(\theta) = \sqrt{\theta} - \theta^2. + Find the maximum likelihood estimation of \theta. +

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