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1. DISTRIBUTION THEORY which is the MGF of X 1 ∼ N r1 (µ 1 , Σ 11 ). Similarly for X 2 . This means that all marginal distributions of a multivariate normal distribution are multivariate normal themselves. Note though, that in general the opposite implication is not true: there are examples of non-normal multivariate distributions who marginal distributions are normal. 1.10.2 Independence and normality We have seen previously that X, Y independent ⇒ Cov(X, Y ) = 0, but not vice-versa. An exception is when the data are (jointly) normally distributed. In particular, if (X, Y ) have the bivariate normal distribution, then Cov(X, Y ) = 0 ⇐⇒ X, Y are independent. Theorem. 1.10.6 Suppose X 1 , X 2 , . . . , X r have a multivariate normal distribution. Then X 1 , X 2 , . . . , X r are independent if and only if Cov(X i , X j ) = 0 for all i ≠ j. Proof. (=⇒)X 1 , . . . , X r independent =⇒ Cov(X i , X j ) = 0 for i ≠ j, has already been proved. (⇐=) Suppose Cov(X i , X j ) = 0 for i ≠ j 65
=⇒ Var(X) = Σ = =⇒ Σ −1 = =⇒ f X (x) = = = ⎡ ⎤ σ 11 0 . . . 0 0 σ 22 . . . . ⎢ ⎣ . . .. . ⎥ .. . ⎦ 0 . . . 0 σ rr ⎡ σ −1 ⎤ 11 0 . . . 0 0 σ −1 22 . . . . ⎢ ⎣ . . .. . ⎥ .. . ⎦ 0 . . . 0 σrr −1 1 (2π) r/2 |Σ| 1/2 e− 1 2 (x−µ)T Σ −1 (x−µ) 1. DISTRIBUTION THEORY and |Σ| = σ 11 σ 22 . . . σ rr 1 1 P r (x i −µ i ) 2 (2π) r/2 (σ 11 σ 22 . . . σ rr ) 1/2 e− 2 i=1 σ ii ( ) ( 1 √ √ e − 1 (x 1 −µ 1 ) 2 1 2 σ 11 √ √ e − 1 2 2π σ11 2π σ22 ) (x 2 −µ 2 ) 2 σ 22 . . . ( ) 1 . . . √ √ e − 1 (xr−µr) 2 2 σrr 2π σrr = f 1 (x 1 )f 2 (x 2 ) . . . f r (x r ) =⇒ X 1 , . . . , X r are independent. Note: (x − µ) T Σ −1 (x − µ) = (x 1 − µ 1 x 2 − µ 2 . . . x r − µ r ) = ⎡ σ −1 ⎤ ⎡ ⎤ 11 0 . . . 0 x 1 − µ 1 0 σ22 −1 . . . . x 2 − µ 2 × ⎢ ⎥ ⎢ ⎥ ⎣ . .. . .. . . ⎦ ⎣ . ⎦ 0 . . . 0 σrr −1 x r − µ r r∑ (x i − µ i ) 2 . σ ii i=1 66
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MATHEMATICAL STATISTICS III Lecture
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1.9 Moments . . . . . . . . . . . .
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1. DISTRIBUTION THEORY ⎧∑ h(x)p
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n(1 − p) E(X) = , p n(1 − p) Va
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1. DISTRIBUTION THEORY ) ( N ) p(x)
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1. DISTRIBUTION THEORY that is, ⎧
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1. DISTRIBUTION THEORY 3. Gamma (K,
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1. DISTRIBUTION THEORY Figure 9: Ca
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Page 23 and 24: 1. DISTRIBUTION THEORY where φ(a)
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Page 35 and 36: ⎧ ⎪⎨ 1 0 < x 2 < 1 f 2 (x 2 )
Page 37 and 38: 1. DISTRIBUTION THEORY Just substit
Page 39 and 40: 1. DISTRIBUTION THEORY 2. Suppose Z
Page 41 and 42: 1. DISTRIBUTION THEORY Solution. St
Page 43 and 44: 1. DISTRIBUTION THEORY Recall stand
Page 45 and 46: ∂ ∂r g 1(r, θ) = ∂ r cos θ
Page 47 and 48: 1. DISTRIBUTION THEORY ⇒ det(G) =
Page 49 and 50: 1. DISTRIBUTION THEORY Definition.
Page 51 and 52: 1. DISTRIBUTION THEORY Proof. Consi
Page 53 and 54: 1. DISTRIBUTION THEORY Proof. 1. (C
Page 55 and 56: 1. DISTRIBUTION THEORY Proof. M X (
Page 57 and 58: 1. DISTRIBUTION THEORY Proof. M AX+
Page 59 and 60: 1. DISTRIBUTION THEORY Remark This
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Page 67: 1. DISTRIBUTION THEORY If we take Z
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Page 73 and 74: 1. DISTRIBUTION THEORY and (R 1 ) n
Page 75 and 76: 1. DISTRIBUTION THEORY Figure 16: D
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Page 79 and 80: 1. DISTRIBUTION THEORY Hence we hav
Page 81 and 82: 2. STATISTICAL INFERENCE Definition
Page 83 and 84: 2. STATISTICAL INFERENCE Figure 19:
Page 85 and 86: 2. STATISTICAL INFERENCE Remark If
Page 87 and 88: 2. STATISTICAL INFERENCE Finally ob
Page 89 and 90: 2. STATISTICAL INFERENCE Proof. (1)
Page 91 and 92: 2. STATISTICAL INFERENCE Hence the
Page 93 and 94: 2. STATISTICAL INFERENCE =⇒ f is
Page 95 and 96: 2. STATISTICAL INFERENCE Examples (
Page 97 and 98: 2. STATISTICAL INFERENCE According
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Page 101 and 102: 2. STATISTICAL INFERENCE To find ˆ
Page 103 and 104: 2. STATISTICAL INFERENCE If X ∼Ex
Page 105 and 106: 2. STATISTICAL INFERENCE (2) f is s
Page 107 and 108: 2. STATISTICAL INFERENCE Then, U n
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Page 111 and 112: 2. STATISTICAL INFERENCE =⇒ W =
Page 113 and 114: 2. STATISTICAL INFERENCE Recall Wal
Page 115 and 116: 2. STATISTICAL INFERENCE Let C 1 =
Page 117 and 118: 2. STATISTICAL INFERENCE Figure 23:
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