Recent content by WMDhamnekar

  1. WMDhamnekar

    I Computing the expectation of the minimum difference between the 0th i.i.d.r.v. and ith i.i.d.r.v.s where 1 ≤ i ≤ n

    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...
  2. WMDhamnekar

    I How to obtain moment bound from the importance sampling identity?

    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...
  3. WMDhamnekar

    I How to obtain moment bound from the importance sampling identity?

    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...
  4. WMDhamnekar

    Chernoff Bounds using importance sampling identity

    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...
  5. WMDhamnekar

    I Chernoff Bounds for Independent Bernoulli Sums

    I cleared my doubt taking suitable guidelines from other statistician on Internet.
  6. WMDhamnekar

    I Basic Measure Theory: Borel ##\sigma##- algebra

    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...
  7. WMDhamnekar

    Chemistry Valence Bond Theory: Energy of a system with H and Cl atoms

    My signature contains the formulas on Ito Calculus. [Not any more; it has been deleted by the Mentors. Please see your PMs]
  8. WMDhamnekar

    Chemistry Valence Bond Theory: Energy of a system with H and Cl atoms

    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...
  9. WMDhamnekar

    I Expected number of random variables that must be observed

    Answer to (a) given by author is correct. My answer is wrong. Thanks for bringing my error to my notice.
  10. WMDhamnekar

    I How to determine if a set is a semiring or a ring?

    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...
  11. WMDhamnekar

    I Expected number of random variables that must be observed

    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?
  12. WMDhamnekar

    I Show that ##Y_{\infty}=0 ##

    Sorry, I forgot to provide one additional information. For ##q =\displaystyle\frac{1}{e+1}, Y_n## is a martingale.
  13. WMDhamnekar

    I Show that ##Y_{\infty}=0 ##

    I have written question also in LaTeX form now.
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