Suppose X is a real valued random variable with μ = E(X) = 0 (a) Show that for any t > 0 and a ≤ x ≤ b, E(e¹x) ≤ es(t(b-a)), where g (u) = log(1 − y + ye") - yu, with y = a/(a - b). (Hint: write X as a convex combination of a and b, where the convex weighting parameter depends upon X. Exploit the convexity of the function x→ ex and the fact that inequalities are preserved upon taking expectations of both sides, since expectations are integrals.) (b) By using Taylor's theorem, show that for all u > 0, Furthermore show that g(u) ≤ u² 8 1²(b-a)² 8 E(e¹x) ≤ e

Suppose X is a real valued random variable with μ = E(X) = 0 (a) Show that for any t > 0 and a ≤ x ≤ b, E(e¹x) ≤ es(t(b-a)), where g (u) = log(1 − y + ye") - yu, with y = a/(a - b). (Hint: write X as a convex combination of a and b, where the convex weighting parameter depends upon X. Exploit the convexity of the function x→ ex and the fact that inequalities are preserved upon taking expectations of both sides, since expectations are integrals.) (b) By using Taylor's theorem, show that for all u > 0, Furthermore show that g(u) ≤ u² 8 1²(b-a)² 8 E(e¹x) ≤ e

Calculus For The Life Sciences

2nd Edition

ISBN:9780321964038

Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Chapter4: Calculating The Derivative

Section4.4: Derivatives Of Exponential Functions

Problem 53E

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