Probability Question

(See picture for better formatting)

Let X be a standard exponential random variable, so for any non-negative Borel function f : R → R,

E(f(X)) = $\int_{0}^{\infty } f(x)\cdot e^{-x} dx$. Let φn : R → R be given by φn(x) = $\cos(\frac{x}{n}) \cdot(1+\frac{x}{n^2})^n$ for all x ∈ R. Show that lim n→∞ E(φn(X)) = 1.

Hint: for all real numbers x, $1 + x $ ≤ $e^x$ by convexity of the exponential function.

Probability Integrals Real Analysis
L Ellis L Ellis
369
Matchmaticians Probability Question File #1 File #1 (PNG)
Report

Answer

Answers can only be viewed under the following conditions:
  1. The questioner was satisfied with and accepted the answer, or
  2. The answer was evaluated as being 100% correct by the judge.
View the answer

1 Attachment

Erdos Erdos
4.8K
  • Erdos Erdos
    0

    You should set later deadlines for your questions. A few hours is too early. Ideally you wanna give at least 24 hours if possible.

  • L Ellis L Ellis
    0

    I agree, but I'm in a bit of a crunch unfortunately!

  • Erdos Erdos
    0

    I am busy at the moment, but If you extend the deadline of your other question I may be able to help. You may also want to offer higher bounties to give users more incentive to accept.

  • L Ellis L Ellis
    0

    I can't extend unfortunately, the deadline is quite soon

  • L Ellis L Ellis
    0

    Thanks for the rest, though!

The answer is accepted.
Join Matchmaticians Affiliate Marketing Program to earn up to a 50% commission on every question that your affiliated users ask or answer.