8Probability and Moment Generating Functions
GX(t) = E tX |
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MX(t) = GX(et) = E eXt = E " |
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P [X = 0] = GX(0)P [X = 1] = G0X(0)
P [X = i] = G(Xi)(0) i!
E [X] = G0X(1 )
E Xk = MX(k)(0)
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9Multivariate Distributions
9.1Standard Bivariate Normal
Let X; Y N (0; 1) ^ X ?? Z where Y = X + p |
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9.2Bivariate Normal
Let X N x; x2 and Y N y; y2 . |
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Conditional mean and variance
E [X j Y ] = E [X] + X (Y E [Y ])
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V [X j Y ] = X 1 2
9.3Multivariate Normal
Covariance matrix (Precision matrix 1) |
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Properties
Z N (0; 1) ^ X = + 1=2Z =) X N ( ; )
X N ( ; ) =) 1=2(X ) N (0; 1)
X N ( ; ) =) AX N A ; A AT
X N ( ; ) ^ kak = k =) aT X N aT ; aT a
10 Convergence
Let fX1; X2; : : :g be a sequence of rv's and let X be another rv. Let Fn denote the cdf of Xn and let F denote the cdf of X.
Types of convergence
D
1. In distribution (weakly, in law): Xn ! X
lim Fn(t) = F (t) 8t where F continuous
n!1
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2. In probability: Xn ! X
(8" > 0) lim P [jXn Xj > "] = 0
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3. Almost surely (strongly): Xn ! X
P hn!1 Xn = Xi |
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