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
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Help asap!
Transcribed Image Text: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
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 2 steps with 2 images
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
