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.
L Ellis
Answer
- The questioner was satisfied with and accepted the answer, or
- The answer was evaluated as being 100% correct by the judge.
1 Attachment
Erdos
-
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!
- answered
- 1302 views
- $8.00
Related Questions
- Finding a unique structure of the domain of a function that gives a unique intuitive average?
- Let $f\in C (\mathbb{R})$ and $f_n=\frac{1}{n}\sum\limits_{k=0}^{n-1} f(x+\frac{k}{n})$. Prove that $f_n$ converges uniformly on every finite interval.
- Operational Research probabilistic models
- Find the arc length of $f(x)=x^{\frac{3}{2}}$ from $x=0$ to $x=1$.
- Evaluate the integral $\int_{-\infty}^{+\infty}e^{-x^2}dx$
- Assume there is no $x ∈ R$ such that $f(x) = f'(x) = 0$. Show that $$S =\{x: 0≤x≤1,f(x)=0\}$$ is finite.
- Determine which one of the following statements is true and explain why
- Probability of picking a red ball
