Intereting Posts

Fundamental Theorem of Calculus: Why Doesn't the Integral Depend on Lower Bound?
Maximum likelihood estimation when the density is $f(x;\theta) = \theta x^{\theta -1} $
Sequence of continuous functions converges uniformly. Does it imply the limit function is continuous?
Gradient-descent algorithm always converges to the closest local optima?
Solving a 2nd order differential equation by the Frobenius method
Minimum Modulus Principle for a constant fuction in a simple closed curve
Why does $\int_a^b fg\, dx = 0$ imply that $f = 0$?
Inverse Matrices and Infinite Series
Writing up a rigorous solution for finding a basis for the $n \times n$ symmetric matrices.
Are there situations outside of set theory where it would be useful if $\mathrm{ICF}$ were true?
If A and B are ideals of a ring, show that A + B = $\{a+b|a \in A, b \in B\}$ is an ideal
A question on measurability in product spaces
What is the order of the sum of log x?
Linear algebra proof regarding matrices
Isometric <=> Left Inverse Adjoint

Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space, and let $X$ be a nonnegative random defined on this space. For any $A\in \mathcal{F}$, let $\mathbb{E}[X;A]:=\mathbb{E}[X\mathbb{1}_{A}]$, where $\mathbb{1}_{A}$ denotes the indicator random variable on $A$. Without assuming $\mathbb{E}\left[\frac{1}{X}\right]<\infty$, show that

\begin{eqnarray}

\lim\limits_{y\downarrow 0} ~~y~\mathbb{E}\left[\frac{1}{X};X>y\right]=0.

\end{eqnarray}

Any initial ideas will be greatly appreciated.

- Probability of men and women sitting at a table alternately
- Sum of Cauchy distributed random variables
- How to calculate $E, n \geq 2$
- Cauchy Schwarz inequality for random vectors
- Can you determine a formula for this problem?
- Probability the three points on a circle will be on the same semi-circle
- Coupon Collector Problem - expected number of draws for some coupon to be drawn twice
- Expected number of tosses to get 3 consecutive Heads
- Change of measure of conditional expectation
- Help with a specific limit $\left( \dfrac{n-1}{n} \right)^n$ as $n \rightarrow \infty$

**Hint**: For any $\epsilon>0$ there exists $y_0>0$ such that $\mathbb{P}(0<X\le y_0)\le\epsilon$ (why?). For $y<y_0$, we have

\begin{align} y\mathbb{E}\left[\frac{1}{X};X>y\right] &= y\left(\mathbb{E}\left[\frac{1}{X};X>y_0\right]+\mathbb{E}\left[\frac{1}{X};y<X\le y_0\right]\right) \\

&\le y\left(\frac{1}{y_0}+\frac{1}{y}\mathbb{P}(y<X\le y_0)\right) \\

&\le \frac{y}{y_0}+\epsilon.

\end{align}

What can you say as $y\rightarrow 0$?

- Number of terms in the expansion of $\left(1+\frac{1}{x}+\frac{1}{x^2}\right)^n$
- Value of Summation of $\log(n)$
- Proof that the preimage of generated $\sigma$-algebra is the same as the generated $\sigma$-algebra of preimage.
- Solving a recursion relation: $a_{n+1}=-3a_n+4a_{n-1}+2$
- Question about primitive roots of p and $p^2$
- A formula for the roots of a solvable polynomial
- Let $f:(\mathbb{R}\setminus\mathbb{Q})\cap \to \mathbb{Q}\cap $. Prove there exists a continuous$f$.
- Associates in Integral Domain
- Contraction and Fixed Point
- Frullani 's theorem in a complex context.
- The ideal generated by a non-compact operator
- Characterising Continuous functions
- Hilbert's Original Proof of the Nullstellensatz
- Birthday Probability
- Is homology with coefficients in a field isomorphic to cohomology?