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1. DISTRIBUTION THEORY P (Y = 0) = P (X = 0) + P (X = 1) + P (X = 2) + P (X = 3) + P (X = 4) = e −λ + e −λ λ + e−λ λ 2 2! + e−λ λ 3 3! + e−λ λ 4 ; 4! P (Y = 10) = P (X = 5) + P (X = 6) + P (X = 7) + P (X = 8) + . . . + P (X = 14), andsoon. 2. Continuous transformation of single RV Let h(X) = aX + b, a ≠ 0. To find h −1 (y) we solve for x in the equation y = h(x), i.e., y = ax + b ⇒ x = y − b a ⇒ h −1 (y) = y − b a = y a − b a Specifically: Hence, f Y (y) = 1 |a| f X ⇒ h −1 (y) ′ = 1 a . ( y − b 1. Suppose Z ∼ N(0, 1) and let X = µ + σZ, σ > 0. Recall that Z has PDF: Hence, f X (x) = 1 ( ) x − µ |σ| φ = σ a ) . φ(z) = 1 √ 2π e −z2 /2 . 2. Suppose X ∼ N(µ, σ 2 ) and let Z = X − µ . σ Find the PDF of Z. 1 √ e − (x−µ)2 2σ 2 , which is N(µ, σ 2 ) PDF. 2π(σ) 15
1. DISTRIBUTION THEORY Solution. Observe that Z = h(X) = 1 σ X − µ σ , ( z + µ ) σ ⇒ f Z (z) = σf X 1 = σf X (µ + σz) σ 1 = σ √ e (−1) 2πσ 2 (2σ 2 ) (µ+σz−µ)2 = 1 √ 2π e (−1) (2σ 2 ) σ2 z 2 = 1 √ 2π e −z2 /2 = φ(z), i.e., Z ∼ N(0, 1). 3. Suppose X ∼ Gamma (α, 1) and let Y = X λ , λ > 0. Find the PDF of Y . Solution. Since Y = 1 X, is a linear function, we have λ f Y (y) = λf X (λy) which is the Gamma(α, λ) PDF. = λ 1 Γ(α) (λy)α−1 e −λy = λα Γ(α) yα−1 e −λy , 1.4 CDF transformation Suppose X is a continuous RV with CDF F X (x), which is increasing over the range of X. If U = F X (x), then U ∼ U(0, 1). 16
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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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