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Contents 1 Distribution Theory 1 1.1 Discrete distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.1.1 Bernoulli distribution . . . . . . . . . . . . . . . . . . . . . . . . 2 1.1.2 Binomial distribution . . . . . . . . . . . . . . . . . . . . . . . . 3 1.1.3 Geometric distribution . . . . . . . . . . . . . . . . . . . . . . . 3 1.1.4 Negative Binomial distribution . . . . . . . . . . . . . . . . . . 3 1.1.5 Poisson distribution . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.1.6 Hypergeometric distribution . . . . . . . . . . . . . . . . . . . . 5 1.2 Continuous Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.2.1 Uniform Distribution . . . . . . . . . . . . . . . . . . . . . . . . 7 1.2.2 Exponential Distribution . . . . . . . . . . . . . . . . . . . . . . 8 1.2.3 Gamma distribution . . . . . . . . . . . . . . . . . . . . . . . . 9 1.2.4 Beta density function . . . . . . . . . . . . . . . . . . . . . . . . 10 1.2.5 Normal distribution . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.2.6 Cauchy distribution . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.3 Transformations of a single RV . . . . . . . . . . . . . . . . . . . . . . 12 1.4 CDF transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 1.5 Non-monotonic transformations . . . . . . . . . . . . . . . . . . . . . . 17 1.6 Moments of transformed RVS . . . . . . . . . . . . . . . . . . . . . . . 18 1.7 Multivariate distributions . . . . . . . . . . . . . . . . . . . . . . . . . 23 1.7.1 Trinomial distribution . . . . . . . . . . . . . . . . . . . . . . . 23 1.7.2 Multinomial distribution . . . . . . . . . . . . . . . . . . . . . . 23 1.7.3 Marginal and conditional distributions . . . . . . . . . . . . . . 24 1.7.4 Continuous multivariate distributions . . . . . . . . . . . . . . . 27 1.8 Transformations of several RVs . . . . . . . . . . . . . . . . . . . . . . 33 1.8.1 Method of regular transformations . . . . . . . . . . . . . . . . 41
1.9 Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 1.9.1 Moment generating functions . . . . . . . . . . . . . . . . . . . 50 1.9.2 Marginal distributions and the MGF . . . . . . . . . . . . . . . 53 1.9.3 Vector notation . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 1.9.4 Properties of variance matrices . . . . . . . . . . . . . . . . . . 56 1.10 The multivariable normal distribution . . . . . . . . . . . . . . . . . . . 57 1.10.1 The multivariate normal MGF . . . . . . . . . . . . . . . . . . . 64 1.10.2 Independence and normality . . . . . . . . . . . . . . . . . . . . 65 1.11 Limit Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 1.11.1 Convergence of random variables . . . . . . . . . . . . . . . . . 71 2 Statistical Inference 77 2.1 Basic definitions and terminology . . . . . . . . . . . . . . . . . . . . . 77 2.2 Minimum Variance Unbiased Estimation . . . . . . . . . . . . . . . . . 81 2.3 Methods Of Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 2.3.1 Method Of Moments . . . . . . . . . . . . . . . . . . . . . . . . 94 2.3.2 Maximum Likelihood Estimation . . . . . . . . . . . . . . . . . 97 2.4 Hypothesis Tests and Confidence Intervals . . . . . . . . . . . . . . . . 105
Page 1: MATHEMATICAL STATISTICS III Lecture
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 and 10: 1. DISTRIBUTION THEORY ) ( N ) p(x)
Page 11 and 12: 1. DISTRIBUTION THEORY that is, ⎧
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
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1. DISTRIBUTION THEORY Proof. 1. (C
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1. DISTRIBUTION THEORY Proof. M X (
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1. DISTRIBUTION THEORY Proof. M AX+
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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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