[SOLVED] Stat435 Homework5

Problem 1: Consider a classification problem with a binary class label Y and a single continuous feature X that takes values in (−4,−2) ∪ (2,4). Suppose (X,Y ) is generated by choosing Y at random with P(Y = 1) = P(Y = 2) = 1/2, and then drawing X conditional on Y according to uniform distributions. Specifically, assume that the class-conditional densities for X are

)      and

In the below we consider 0-1 loss, that is, the risk of a classifier is the probability of an error.

•  What is the marginal distribution of X? What is the conditional distribution of Y given X?
•  What is the Bayes rule f_B(x) and its risk P(Y 6= f_B(x)?

Explain!

• Let f^ˆ₁(x;S) be the 1-nearest neighbor classifier based on a training sample S = {(x₁,y₁),…,(x_n,y_n)} of i.i.d observations of (X,Y ). What is the risk Pr(Y 6= f^ˆ₁(X;S)? Explain. (Here, the risk is computed by integrating over training data and a new independent pair (X,Y )).
•  Under the same scenario calculate the risk of the 3-nearest neighbor classifier.
•  Which method, 1-nearest neighbor or 3-nearest neighbor, has smaller risk in this problem?

2. ISLR Section 8.4 Problem 3 
3. ISLR Section 8.4 Problem 9 (a) …(g)

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