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 Dec20 awarded Yearling Sep24 awarded Autobiographer Dec20 awarded Yearling Sep24 comment Show the mapping is $C^1$ Oh ok, I see that now. But then why is the sup-norm important? Is it simply because it makes C([0, 1]) a Banach space? Sep23 asked Show the mapping is $C^1$ May10 answered equivalency of weak and strong convergence May10 comment How to prove a limit with a recurrence? you might look here: math.stackexchange.com/questions/234814/… May9 comment How I can find the expected value of $G$? have you read the wiki page on the negative binomial? It explains there what you would plug in and tells you the pmf, from which you can calculate expectation. May8 answered Find the volume of a regular Region in $\mathbb {R^{3}}$ May8 answered Ratios/Proportions of numbers: problem solving Apr8 answered Differentiation confusion Apr8 comment Proof for two sequences producing the maximum value when sorted I assume you actually mean $x_1 y_1 + x_2 y_2 + \ldots + x_n y_n$? And your contradictory assumption needs modification. 1) Just because you assume $S$ is not a maximum does not mean that $S$ is now a minimum. 2) You don't want to change the given, which is that $x_1, x_2, \ldots, x_n$ and $y_1, y_2, \ldots y_n$ are sorted in non decreasing order. Feb23 awarded Commentator Feb23 accepted The Legendre Transform of Bernoulli r.v.s Feb23 comment The Legendre Transform of Bernoulli r.v.s Ah, that makes sense. One more question: so I got $\alpha_\beta = \log(\frac{\beta(1 - p)}{p(1 - \beta)})$. How do I calculate $H(\alpha_\beta) = -\log E(\frac{\beta (1 - p)}{p (1 - \beta)})^X = \log E(\frac{p (1 - \beta)}{\beta (1 - p)})^X$? Based on the statement that should give $\log(\frac{1 - \beta}{1 - p})$ for $\beta \in [0, 1]$, but I don't know how to calculate that. Feb23 comment Legendre transform of log moment function $e^{\lambda \text{ esssup } X} \geq Ee^{\lambda X}$, so take the logarithms of both sides, then divide by $\lambda$, and take the limit as $\lambda \rightarrow +\infty$. Since $\frac{\Lambda(\lambda)}{\lambda} = \Lambda'(\lambda)$ you're done. Feb22 asked The Legendre Transform of Bernoulli r.v.s Feb22 answered Legendre transform of log moment function Feb6 comment Maclaurin expansion of $\sqrt{\cos 2x}$ and $\tan^2 x$ up to degree 4 Not really, if you only need up to degree 4 then squaring will give you the 4 terms with relative ease. Feb5 comment Maclaurin expansion of $\sqrt{\cos 2x}$ and $\tan^2 x$ up to degree 4 Are you sure they want you to find the Maclaurin expansion of $\sqrt{\cos(2x)}$? It really is just horrible differentiation with no visible pattern, but it doesn't seem like a very useful exercise if that is the case. The one for $\tan^2(x)$ can be found more easily by simply squaring the Maclaurin expansion for $\tan(x)$.