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Unformatted text preview: random variables. A common case is
when we have diﬀerent probability densities for diﬀerent classes (ωj ) P (ωj x) = p(xωj )P (ωj )
p (x) • P (ωj ) = prior probability of ωj
• p(x) = evidence
• P (ωj x) = posterior probability of ωj
• p(xωj ) = likelihood of ωj with respect to x 40 Expected Value Expected value or mean E (X ) = xp(x) What is the expected value of playing a game where there is a 10% chance of
winning $10 and a 80% chance of losing $1 and a 10% chance of losing $2? 41 Expected Value Expected value or mean E (X ) = xp(x) What is the expected value of playing a game where there is a 10% chance of
winning $10 and a 80% chance of losing $1 and a 10% chance of losing $2?
E (X ) = .1 ∗ 10 − .8 ∗ 1 − .1 ∗ 2 = 0 The expected value (expected payoﬀ) is $0. 42 Expected Value Expected value or mean E (X ) = E (X ) = xp(x) xp(x)dx 43 Expected Value Expected value or mean E (X ) = E (X ) = xp(x) xp(x)dx What is the expected value (mean) of a uniform distribution from 0 to 2.
First note a uniform distribution from 0 to 2 would have p(x)=0.5 for x ∈ [0, 2]
2
thus E (X ) = 0 x(0.5)dx = 0.25x22 = 1
0 44 45 Variance V ar(X ) = E (X − E (X ))2 = V ar(X ) = P (X )(X − E (X ))2 (x − E (x))2p(x)dx • It has the maximum entropy of all distributions with a given mean and variance
• It’s been well studied
• It’s analytically tractable!
• Central Limit Theorem: sum of a large number of independent random variables
is normally distributed applet
Pattern Densities are commonly modeled by
Normal Densities for several reasons
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¨ The Normal Density 46 Univariate normal density (x−µ)2
1
−
p (x) = √
e 2σ 2
2πσ has mean = µ
variance = σ 2
has roughly 95% of its area within 2 standard deviations on either side of the
mean (this is relevant for ttests). 47 Standard Normal Gaussian with mean 0 and variance 1 (µ = 1, σ 2 = 1)
1 − x2
√ e2
2π 48 Univariate normal/Gaussian density What is ? ∞ 1 −(x−...
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This note was uploaded on 02/09/2014 for the course COGS 109 taught by Professor Staff during the Fall '08 term at UCSD.
 Fall '08
 staff

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