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1. DISTRIBUTION THEORY Proof. F U (u) = P (U ≤ u) = P {F X (X) ≤ u} = P {X ≤ F −1 X (u)} = F X {F −1 X (u)} = u, for 0 of U(0, 1), so the result is proved. The converse also applies. If U ∼ U(0, 1) and F is any strictly increasing “on its range” CDF, then X = F −1 (U) has CDF F (x), i.e., F X (x) = P (X ≤ x) = P {F −1 (U) ≤ x} = P (U ≤ F (x)) = F (x), as required. 1.5 Non-monotonic transformations Theorem 1.3.2 applies to monotonic (either strictly increasing or decreasing) transformations of a continuous RV. In general, if h(x) is not monotonic, then h(x) may not even be continuous. For example, integer part of x ↗ if h(x) = [x], then possible values for Y = h(X) are the integers ⇒ Y is discrete. However, it can be shown that h(X) is continuous if X is continuous and h(x) is piecewise monotonic. 17
1. DISTRIBUTION THEORY 1.6 Moments of transformed RVS Suppose X is a RV and let Y = h(X). If we want to find E(Y ) , we can proceed as follows: 1. Find the distribution of Y = h(X) using preceeding methods. ⎧∑ yp(y) Y discrete ⎪⎨ y 2. Find E(Y ) = ⎪⎩ Theorem. 1.6.1 ∫ ∞ −∞ yf(y) dy Y continuous (forget X ever existed!) If X is a RV of either discrete or continuous type and h(x) is any transformation (not necessarily monotonic), then E{h(X)} (provided it exists) is given by: ⎧∑ h(x) p(x) X discrete ⎪⎨ x E{h(X)} = ⎪⎩ Proof. ∫ ∞ −∞ h(x)f(x) dx X continuous Not examinable. Examples: 1. CDF transformation Suppose U ∼ U(0, 1). How can we transform U to get an Exp(λ) RV Solution. Take X = F −1 (U), where F is the Exp(λ) CDF. Recall F (x) = 1 − e −λx for the Exp(λ) distribution. To find F −1 (u), we solve for x in F (x) = u, 18
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Page 3 and 4: 1.9 Moments . . . . . . . . . . . .
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Page 7 and 8: n(1 − p) E(X) = , p n(1 − p) Va
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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 Let C 1 =
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2. STATISTICAL INFERENCE Figure 23:
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