sos440
Reputation
39,947
884/400 score
 1d awarded Nice Answer 2d revised Average distance between two randomly chosen points in unit square (without calculus) added 808 characters in body 2d answered Average distance between two randomly chosen points in unit square (without calculus) 2d revised Independence of Random Variables LaTeXify 2d comment Independence of Random Variables Hint. Notice that the $\sigma$-algebras satisfy $\sigma(f(X)) \subseteq \sigma(X)$ and $\sigma(g(Y)) \subseteq \sigma(Y)$. Also, what is the complement of a random variable? May21 revised prove that $f(z)+f(iz)=0$ please added 538 characters in body May21 answered prove that $f(z)+f(iz)=0$ please May16 answered If $f:[0,1]\to\mathbf{R}$ is continuous, and that $\int_0^1 f(x)x^k \, dx = 0$ for all integers $k>2004$, show that $f = 0$. May16 comment If $f:[0,1]\to\mathbf{R}$ is continuous, and that $\int_0^1 f(x)x^k \, dx = 0$ for all integers $k>2004$, show that $f = 0$. How about showing that $\tilde{f}(x) = x^{2005}f(x)$ is zero? May16 comment Integral of $\int_{-\infty}^\infty\frac{e^{-y^2/2} \sinh(cy)^2} {\cosh(Mcy) }dy$ @Boby, That's right. It is change of variable. Also, $\sec$ is correct. For example, the integral diverges to $\infty$ as $M \to 2^+$, and this phenomenon is exactly replicated by the answer. May15 answered Integral of $\int_{-\infty}^\infty\frac{e^{-y^2/2} \sinh(cy)^2} {\cosh(Mcy) }dy$ May14 comment How to simplify this summation, or express as integral? $\sum_{t=-\infty}^\infty\frac1{\sqrt{(t+ax)^2+4}}$ Of course, you can always write it as $+\infty$. May13 awarded Nice Answer May12 awarded Good Answer May7 comment What is the difference between weak and strong law of large numbers? My guess is that (1) SLLN is much harder to prove than WLLN (particularly in preliminary probability courses), and (2) they generalize in a slightly different way. Indeed, there exists an i.i.d. sequence $(X_n)$ of random variables for which WLLN holds but SLLN does not hold. May7 comment Is there a name for the relation $\int^b_a f(x)\ dx=\int_a^c f(x)\ dx+\int_c^b f(x)\ dx$? I would call it additivity, since it generalizes to the additivity of the signed measure $E \mapsto \int_E f(x) \, dx$ with integrable $f$ (though this covers only the case $a\leq c\leq b$). But many preliminary analysis textbooks just seem to leave it unnamed. May6 awarded Necromancer May4 awarded Good Answer May4 comment On the inequality $\int_{-\infty}^{+\infty}\frac{(p'(x))^2}{(p'(x))^2+(p(x))^2}\,dx \le n^{3/2}\pi.$ @DavidSpeyer, Thank you for the choosing my answer! May4 revised Computing $\int_{0}^{\infty} \frac{x^2 + a}{x^6 + a^3}dx$ added 2 characters in body