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1 I am trying to prove the following inequality: Given a random variable X with 0 ≤ X ≤ 1 E [ e t X] ≤ ( 1 − E [ X]) + E [ X] e t, t > 0 I have the following steps so far: Since X ∈ [ 0, 1] E [ X] ∈ [ 0, 1] Define a convex function f ( X) = e t X Then

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1 e t x ≤ ( 1 − x) + x e t for 0 ≤ x ≤ 1 and t ≥ 0. To see this fix x and note that equlaity holds when t = 0; now check that the derivative of e t x − ( 1 − x) + x e tH-E-B is situated at the main intersection of North Greenville Avenue and East Exchange Parkway, in Allen, Texas. By car . Merely a 1 minute trip from High Meadow Drive, Meadowgate Drive, Roaring Springs Drive or Exit 37 (Central Expressway) of US-75; a 5 minute drive from Central Expressway (US-75), East Stacy Road and North Allen …

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Feb 9, 2022 · 0. Note that deterministically, e t x ≤ ( 1 − x) + x e t ∀ x ∈ [ 0, 1], t ∈ R ( 1). To see this, consider a discrete random variable Y having two-point support { 0, t } with respective probability masses 1 − x, x. Since z → e z is a convex function, Jensen's inequality tells us.

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