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1. DISTRIBUTION THEORY This is the distribution for the waiting time until the first occurrence in a Poisson process with rate parameter λ. 1. If X ∼ Exp(λ) then, P (X ≥ t + x|X ≥ t) = P (X ≥ x) (memoryless property) 2. It can be obtained as limiting form <strong>of</strong> geometric distribution. 1.2.3 Gamma distribution gamma fct : Γ(α) = mgf : M X (t) = ∫ ∞ 0 t α−1 e −t dt ( ) α λ , t < λ. λ − t Suppose Y 1 , . . . , Y K are independent Exp(λ) RV’s and let X = Y 1 + · · · + Y K . Then X ∼ Gamma(K, λ), for K integer. In general, X ∼ Gamma(α, λ), α > 0. 1. α is the shape parameter, λ is the scale parameter Figure 7: Gamma Distribution Note: if Y ∼ Gamma (α, 1) and X = Y , then X ∼ Gamma (α, λ). That is, λ is λ scale parameter. ( p 2. Gamma 2 , 1 distribution is also called χ 2) 2 p (chi-square with p df) distribution if p is integer; χ 2 2 = exponential distribution (for 2 df). 9
1. DISTRIBUTION THEORY 3. Gamma (K, λ) distribution can be interpreted as the waiting time until the K th occurrence in a Poisson process. 1.2.4 Beta density function Suppose Y 1 ∼ Gamma (α, λ), Y 2 ∼ Gamma (β, λ) independently, then, Remark: X = Y 1 Y 1 + Y 2 ∼ B(α, β), 0 ≤ x ≤ 1. Figure 8: Beta Distribution 1.2.5 Normal distribution X ∼ N(µ, σ 2 ); M X (t) = e tµ e t2 σ 2 /2 . 1.2.6 Cauchy distribution Possible values: x ∈ R <strong>PDF</strong>: f(x) = 1 ( ) 1 ; (location parameter θ = 0) π 1 + x 2 10
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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 and 10: 1. DISTRIBUTION THEORY ) ( N ) p(x)
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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)
Page 25 and 26: 1. DISTRIBUTION THEORY Examples 1.
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Page 35 and 36: ⎧ ⎪⎨ 1 0 < x 2 < 1 f 2 (x 2 )
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Page 45 and 46: ∂ ∂r g 1(r, θ) = ∂ r cos θ
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Page 57 and 58: 1. DISTRIBUTION THEORY Proof. M AX+
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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 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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