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1. DISTRIBUTION THEORY and for f X1 (x 1 ) > 0, the conditional PDF is given by f X2 |X 1 (x 2 |x 1 ) = f(x 1, x 2 ) f X1 (x 1 ) . Remarks: 1. f X2 |X 1 (x 2 |x 1 ) cannot be interpreted as the conditional PDF of X 2 |{X 1 = x 1 } because P (X 1 = x 1 ) = 0 for any continuous distribution. Proper interpretation is the limit as δ → 0 in X 2 |X 1 ∈ B(x 1 , δ). 2. f X2 (x 2 ), f X1 |X 2 (x 1 |x 2 ) are defined analogously. Definition. 1.7.7 (Independence) Continuous RV’s X 1 , X 2 , . . . , X r are said to be independent if their joint PDF satisfies: f(x 1 , x 2 , . . . , x r ) = f 1 (x 1 )f 2 (x 2 ) . . . f r (x r ), for some functions f 1 , f 2 , . . . , f r and all x 1 , . . . , x r Remarks 1. Easy to check that if X 1 , . . . , X r are independent then each f i (x i ) = c i f Xi (x i ). Moreover c 1 , c 2 , . . . , c r = 1. 2. If X 1 , X 2 , . . . , X r are independent, then it can be checked that: for any partitioning of X = f x1 |x 2 (x 1 |x 2 ) = f X1 (x 1 ) ( X1 X 2 ) . Examples 1. If (X 1 , X 2 ) has the uniform distribution on the unit disk, find: (a) the marginal PDF f X1 (x 1 ) (b) the conditional PDF f X2 |X 1 (x 2 |x 1 ). 29
1. DISTRIBUTION THEORY Solution. Recall ⎧ 1 ⎪⎨ x 2 1 + x 2 2 < 1 π f(x 1 , x 2 ) = ⎪⎩ 0 otherwise (a) f X1 (x 1 ) = = ∫ ∞ −∞ f(x 1 , x 2 ) dx 2 ∫ − √ 1−x 2 1 −∞ = 0 + x 2 π = √ ∣ √ − ∫ √ 1−x 2 1 0 dx 2 + 1−x 2 1 ⎧ 2 √ 1 − x ⎪⎨ 2 1 π 1−x 2 1 + 0 √ − 1−x 2 1 −1 < x 1 < 1 ⎪⎩ 0 otherwise. ∫ 1 ∞ π dx 2 + √ 0 dx 2 1−x 2 1 i.e., a semi-circular distribution (ours has some distortion). Figure 13: A graphic showing the conditional distribution and a semi-circular distribution. 30
Page 1 and 2: MATHEMATICAL STATISTICS III Lecture
Page 3 and 4: 1.9 Moments . . . . . . . . . . . .
Page 5 and 6: 1. DISTRIBUTION THEORY ⎧∑ h(x)p
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Page 17 and 18: 1. DISTRIBUTION THEORY F Y (y) = P
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Page 21 and 22: 1. DISTRIBUTION THEORY 1.6 Moments
Page 23 and 24: 1. DISTRIBUTION THEORY where φ(a)
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Page 75 and 76: 1. DISTRIBUTION THEORY Figure 16: D
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2. STATISTICAL INFERENCE Remark If
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2. STATISTICAL INFERENCE Proof. (1)
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2. STATISTICAL INFERENCE Examples (
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