Properties of the Distribution Function G (Part 2) | Massachusetts Institute of Technology - Edubirdie

University:
Massachusetts Institute of Technology
Course:
15.501/516 | Accounting
Academic year:

2023
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138

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2
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Sorabellaxd8

Advanced Stochastic Processes. LECTURE 4 Properties of the distribution function G We started with a distribution function F (x) and considered a modiﬁed distribution function � x ∗ ∗ −1 G(x) = M (θ ) eθ y dF (y). −∞ ˜ be a random variable We showed that G is indeed a probability distribution function. Let X distributed according to G. Lemma 4.1. The following holds (a) E[X̃] = a. (b) E[X̃ 2 ] < ∞. Proof. By deﬁnition θ∗ = arg sup(aθ − log M (θ)) θ = arg inf(−aθ + log M (θ)) θ = arg inf (e−aθ+log M (θ) ) θ = arg inf (M (θ)e−aθ ) θ = arg inf E[eθ(X1 −a) ] θ Since 0 < θ∗ < θ0 , then the derivative of E[eθ(X1 −a) ] at θ∗ must be equal to zero. Taking derivative at θ∗ , we conclude E[(X1 − a)eθ Multiplying both parts by e θ∗ a ∗ (X −a) 1 ]=0 we obtain E[(X1 − a)eθ ∗X 1 ] = 0, implying E[X1 eθ ∗X 1 ] = aE[eθ 1 ∗X 1 ] = aM (θ∗ ). 2 But � E[X1 e θ∗ X1 ∞ ∗ yeθ y dF (y) −∞ � ∞ ∗ ∗ −1 ∗ = M (θ )M (θ ) yeθ y dF (y) −∞ � ∞ = M (θ∗ ) ydG(y) ]= −∞ ˜1] = M (θ )E[X ∗ ˜ 1 ] = a. Therefore E[X ˜ 2 ] < ∞ will be discussed during the recitation. The second part E[X 1 �

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