Probability Question

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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 ?n : R ? R be given by ?n(x) = $\cos(\frac{x}{n}) \cdot(1+\frac{x}{n^2})^n$ for all x ? R. Show that lim n?? E(?n(X)) = 1.

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

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

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

  • 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.

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

  • Thanks for the rest, though!

The answer is accepted.