Normal Distribution

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Normal Distribution
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Normal Distribution (Gaussian)

• x -50.015
• y normpdf(x,-2,sqrt(0.5))
• plot(x,y,'m')
• Warning normpdf uses N(µ,s)

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Normal Distribution (cont.)
68
cutoffnorminv(0.16 0.84, 0, sqrt(1)) xlo
cutoff(1), x(cutoff(1)ltxltcutoff(2)),
cutoff(2) ylo 0, y(cutoff(1)ltx
xltcutoff(2)), 0 patch(xlo,ylo,'r','FaceAlpha',0.
2')
95
99.7
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Why should we care?
likelihood
prior
posterior
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Why should we care? (cont.)
p1 normpdf(5,4,sqrt(1)) p2
normpdf(5,7,sqrt(0.5))
p1 normpdf(5,4,sqrt(1)) 0.2 p2
normpdf(5,7,sqrt(0.5)) 0.8
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Joint Gaussian Distribution
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Bivariate Gaussian (d2)
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Bivariate Gaussian (d2)
x1 -30.13 x2 -30.13 F mvnpdf(X1()
X2(),mu,sigma) mesh(x1,x2,F) contour(x1,x2,F)
N 1000 y mvnrnd(mu, sigma,
N) plot(y(,1),y(,2), '.')
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Bivariate Gaussian (d2)
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Bivariate Gaussian (d2)
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Why should we care?
load vowels.mat Pa Na/N mu_a mean(a) sigma_a
cov(a) x 400, 1000 logP_ax
log(mvnpdf(x,mu_a,sigma_a)) log(Pa)
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Some Gaussian Tricks
1) Gaussian Marginals are Gaussian
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Some Gaussian Tricks
2) Conditional of Gaussian is Gaussian
curve
area
normalization
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Some Gaussian Tricks
3) Sum of independend Gaussian is Gausian
15
Some Gaussian Tricks
4) Uncorellatedness implies independence (for
joint Gaussian only)
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Some Gaussian Tricks
17
Some Gaussian Tricks
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Central Limit Theorem
Sum of N dies