Name: ———————-
Practice Quiz - 1
CS 591U/791V - Pattern Recognition
Posted on: September 27, 2005
Note:
Univariate normal density: N(µ, σ2)=1
√2πσ eh−1
2(x−µ
σ)2i.
1. [3 points] List the 3 general forms of learning that can be used by a pattern recognition system.
2. Consider a two-category problem where the true state of nature is represented by the variables ω1
and ω2. Let αi(i= 1,2) be the action taken on observing a particular input x. Let λij indicate the
loss incurred when taking action αigiven that the true category is ωj.
(a) [2 points] Write down expressions for the conditional-risks R(α1|x)and R(α2|x).
(b) [4 points] Derive an expression for the likelihood ratio p(x|ω1)/p(x|ω2).
3. [6 points] Let p(x|ωi)∼N(µi, σ2)for a two-category one-dimensional problem with P(ω1) =
P(ω2) = 1/2. Show that the minimum probability of error is given by,
Pe=1
√2πZ∞
a
e−u2
2du,
where a=|µ2−µ1|/2σ.
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