Problem :Let ##X_0,X_1,\dots,X_n## be independent random variables, each distributed uniformly on [0,1].Find ## E\left[ \min_{1\leq i\leq n}\vert X_0 -X_i\vert \right] ##.
Would any member of Physics Forum take efforts to explain with all details the following author's solution to this...
My Answer:
The importance sampling identity states that for any measurable function f and random variable X with probability density function p, the expected value of f(X) can be expressed as:
##E[f(X)] = \int f(x) p(x) dx = \int f(x) \frac{p(x)}{q(x)} q(x) dx,##
where q is another probability...
Let ##X## be a non-negative random variable and let a > 0. We want to bound the probability ##P\{X \geq a\}## in terms of the moments of X.
- Define a function ##h(x) = \mathbb{1}\{x \geq a\}##, where ##\mathbb{1}\{\cdot\}## is the indicator function that returns 1 if the argument is true and 0...
How to use importance sampling identity to obtain the Chernoff bounds as given below?
Let X have moment generating function ##\phi(t)= E[e^{tX}]##. Then, for any c > 0 ,
##P[X\geq c ]\leq e^{-tc} \phi(t), \text{if t > 0}##
##P[X \leq c]\leq e^{-tc}\phi(t), \text{if t<0} ##
Solution...
My answer:
We can proceed as follows. Let ##x \in U_f##. Since ##f## is discontinuous at ##x##, there exists ##\varepsilon > 0## such that for some ##\delta > 0##, we can find ##y, z \in B_{\varepsilon}(x)## with ##d_2(f(y),f(z)) > \delta##. Therefore, ##x \in U^{\delta,\varepsilon}_f##. So...
Answer:
The energy of a system with H and Cl atoms at varying distances can be represented by a curve that shows the potential energy of the system as a function of the distance between the two atoms. At very large distances, the potential energy is zero because there is no interaction between...
Let E be a finite nonempty set and let ## \Omega := E^{\mathbb{N}}##be the set of all E-valued
sequences ##\omega = (\omega_n)_{n\in \mathbb{N}}F##or any ## \omega_1, \dots,\omega_n \in E ## Let
##[\omega_1, \dots,\omega_n]= \{\omega^, \in \Omega : \omega^,_i = \omega_i \forall i =1,\dots,n...
In my opinion, answer to (a) is ## \mathbb{E} [N] = p^{-4}q^{-3} + p^{-2}q^{-1} + 2p^{-1} ##
In answer to (b), XN is wrong. It should be XN=p-4q-3 - p-3 q-2- p-2 q-1 - p-1. This might be a typographical error.
Is my answer to (a) correct?