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1. DISTRIBUTION THEORY Figure 3: p = 1 3 If white ball out → Figure 4: p = 1 2 (without replacement) Figure 5: p = 1 3 (with replacement) Intuitively, this implies that for M, N >> n, the hypergeometric and binomial probabilities should be very similar, and this can be verified for fixed, n, x: 0 lim N 10 M @ A@ M,N→∞ x n − x 0 M M + N → p M + N 1 @ A n 1 A = ( n x) p x (1 − p) n−x . 1.2 Continuous Distributions 1.2.1 Uniform Distribution CDF, for a < x < b: F (x) = ∫ x −∞ f(x) dx = ∫ x a 1 b − a dx = x − a b − a , 7
1. DISTRIBUTION THEORY that is, ⎧ 0 x ≤ a ⎪⎨ x − a F (x) = b − a a < x < b ⎪⎩ 1 b ≤ x Figure 6: Uniform distribution CDF M X (t) = etb − e ta t(b − a) A special case is the U(0, 1) distribution: ⎧ ⎪⎨ 1 0 < x < 1 f(x) = ⎪⎩ 0 otherwise, F (x) = x for 0 < x < 1, E(X) = 1 2 , Var(X) = 1 12 , M(t) = et − 1 . t 1.2.2 Exponential Distribution CDF: F (x) = 1 − e −λx , M X (t) = λ λ − t . 8
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
Page 7 and 8: n(1 − p) E(X) = , p n(1 − p) Va
Page 9: 1. DISTRIBUTION THEORY ) ( N ) p(x)
Page 13 and 14: 1. DISTRIBUTION THEORY 3. Gamma (K,
Page 15 and 16: 1. DISTRIBUTION THEORY Figure 9: Ca
Page 17 and 18: 1. DISTRIBUTION THEORY F Y (y) = P
Page 19 and 20: 1. DISTRIBUTION THEORY Solution. Ob
Page 21 and 22: 1. DISTRIBUTION THEORY 1.6 Moments
Page 23 and 24: 1. DISTRIBUTION THEORY where φ(a)
Page 25 and 26: 1. DISTRIBUTION THEORY Examples 1.
Page 27 and 28: 1. DISTRIBUTION THEORY Possible val
Page 29 and 30: 1. DISTRIBUTION THEORY Remarks 1. O
Page 31 and 32: 1. DISTRIBUTION THEORY Figure 12: A
Page 33 and 34: 1. DISTRIBUTION THEORY Solution. Re
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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1. DISTRIBUTION THEORY 2. If Z 1 ,
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1. DISTRIBUTION THEORY where X i ,
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1. DISTRIBUTION THEORY ⎧ ⎪⎨
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1. DISTRIBUTION THEORY If we take Z
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=⇒ Var(X) = Σ = =⇒ Σ −1 = =
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1. DISTRIBUTION THEORY 2. Independe
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1. DISTRIBUTION THEORY and (R 1 ) n
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1. DISTRIBUTION THEORY Figure 16: D
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1. DISTRIBUTION THEORY Remarks This
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1. DISTRIBUTION THEORY Hence we hav
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2. STATISTICAL INFERENCE Definition
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2. STATISTICAL INFERENCE Figure 19:
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2. STATISTICAL INFERENCE Remark If
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2. STATISTICAL INFERENCE Finally ob
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2. STATISTICAL INFERENCE Proof. (1)
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2. STATISTICAL INFERENCE Hence the
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2. STATISTICAL INFERENCE =⇒ f is
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2. STATISTICAL INFERENCE Examples (
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2. STATISTICAL INFERENCE According
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2. STATISTICAL INFERENCE p = 2 =⇒
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2. STATISTICAL INFERENCE To find ˆ
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2. STATISTICAL INFERENCE If X ∼Ex
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2. STATISTICAL INFERENCE (2) f is s
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2. STATISTICAL INFERENCE Then, U n
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2. STATISTICAL INFERENCE and the al
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2. STATISTICAL INFERENCE =⇒ W =
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2. STATISTICAL INFERENCE Recall Wal
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2. STATISTICAL INFERENCE Let C 1 =
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2. STATISTICAL INFERENCE Figure 23:
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