Recent content by bennyzadir

I would really appreciate if you could help me solving this limit problem! Determine the limit without using L'Hospital's rule! $$ \lim_{x\to -2} \sin(\frac{\pi x}{2})\frac{x^2+1}{x+2} = ?$$ Thank you in advance!

Thank you so much!

How to prove that if $\varphi$ is the characteristic function of an integer valued distribution, then the probability mass function can be computed as $ p(k) = \frac{1}{2\pi} \cdot \int^{\pi}_{-\pi} e^{-ikt}\varphi(t) dt \;,\forall k \in \mathbb{Z} $ I would be really grateful if you could help me.

Thank you so much for your quick and clear answer. I really appreciate it!

Thank you for everybody who replied to my post, special thanks to girdav for the hint and to Evgeny.Makarov for 'protecting' my question. Anyway, I am afriad I still can't solve the problem. To tell the truth I don't really see how the lemma could be used in this case. Which form of the lemma do...

Let be $X_1, X_2, \dots $ independent random variables. My question is how can we show that $sup_n X_n <\infty$ almost surely $ \iff \sum_{n=1}^{\infty} \mathbb{P}(X_n>A)<\infty $ for some positive finite A number. Thank you very much for your help in advance!

Thank you for your answer. Do you have any idea for part b) ?

Let be $X_1, X_2, ..., X_n, ... $ independent identically distributed random variables with mutual distribution $ \mathbb{P}\{X_i=0\}=1-\mathbb{P}\{X_i=1\}=p $. Let be $ Y:= \sum_{n=1}^{\infty}2^{-n}X_n$. a) Prove that if $p=\frac{1}{2}$ then Y is uniformly distributed on interval [0,1]. b) Show...

Thanks for your quick help and for your patient. All the best! Merry Cristmas!

Thank you very much for your detailed answer. I am afraid I don't really understand your notation. Does X. denote derivative, dot products or just a fullstop? Thanks once more!

To tell the truth, I don't really see the point. Curvature line is a synonym of line of curvature. The direction of an eigen vector of the second fundamental form is the same as the direction of the principal curvature. How does the statement follow from the definition?

In my terminology curvature line is a curve on a surface whose tangents are always in the direction of principal curvature. Thank you for your help in advance!

Let be w=w(σ) a curve on a surface parametrized by the arc length (the natural parametrization). Consider the m surface normal along this curve as the function of the σ arc length of the curve. Prove that m'(σ) is parallel to the t(σ) tangent unit vector of the curve for all σ, IFF this curve is...

Thank you very much for your clear and understandable answer.

Let be k \leq n poitive integers. How to show that \left (1+\frac1 n \right)^k \leq 1 + \frac{ke}{n} . It seems to me that it has something to do with Bernoulli's inequality. Thank you in advance!