Description
Question 1 (Problem 5 – Final 2018)
Consider an already-trained Gaussian mixture model (GMM) that is trained to fit data on student
performance in a class. The GMM uses two components (K = 2) as the class consists of two
categories of students: undergraduate students (category 1) and graduate students (category 2).
The learned parameters of the GMM are as follows.
• The weights of the two categories are w1 = 2/3 (undergraduate) and w2 = 1/3 (graduate).
• The distribution that fits scores in category 1 is N (x; 70, 102
).
• The distribution that fits scores in category 2 is N (x; 80, 5
2
).
(a) According to the GMM, what is the probability that an arbitrarily selected student scores
greater than 80%? That is, compute Pr[X ≥ 80], where X denotes the score of the student.
(In your computation, use the approximation that for zero-mean σ
2
-variance random variable
X, i.e., X ∼ N (x; 0, σ2
), Pr[|X| ≤ σ] = 2/3).
(b) If a particular student has a score greater than 80, what is the probability that the student
is from category 1? That is, compute Pr[class = 1|X ≥ 80]. (Use the same approximation as
in the previous part.)
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