Articles of means

If $Y$ is symmetrically distributed around $E(Y)$ then $E(Y^j)=E(Y)^j$ for every odd $j$, why not for $j$ even?

In this statement “If a random variable Y is symmetrically distributed around its mean $\mu_Y$, that is, $Y=\mu_Y+U$, where U is symmetrically distributed aronud zero- then the odd moments of Y are powers of $\mu_Y$, i.e. $E[(Y)^j]= (\mu_Y)^j $ for any odd integer j, assuming the moment exists”. I think this statement is true for […]

Expected value of lognormal distribution.

Hi I’m stuck on this question: Recall that $X$ is said to have a lognormal distribution with parameters $\mu$ and $\sigma^2$ if $\log(X)$ is normal with mean $\mu$ and variance $\sigma^2$. Suppose $X$ is such a lognormal random variable. Find $\mathrm{E}[X]$. Find $\mathrm{Var}(X)$. I know that the approach is to find the moment generating function […]

Prove $\frac{\sqrt{(x+y)(y+z)(z+x)}}{2} \geq \sqrt{\frac{xy+yz+zx}{3}}$ for $x,y,z \geq 0$

We have geometric mean of pairwise arithmetic means on the left, which obeys the following inequality: $$\frac{x+y+z}{3} \geq \color{blue}{ \frac{\sqrt[3]{(x+y)(y+z)(z+x)}}{2} } \geq \sqrt[3]{xyz}$$ And on the right we have root-mean-square of geometric means, obeying the same inequality: $$\frac{x+y+z}{3} \geq \color{blue}{\sqrt{\frac{xy+yz+zx}{3}} } \geq \sqrt[3]{xyz}$$ This time I checked with Wolfram Alpha first, and apparently, the inequality […]

How to go upon proving $\frac{x+y}2 \ge \sqrt{xy}$?

This question already has an answer here: Proving the AM-GM inequality for 2 numbers $\sqrt{xy}\le\frac{x+y}2$ 5 answers How to prove that $\frac{a+b}{2} \geq \sqrt{ab}$ for $a,b>0$? [duplicate] 3 answers

Arithmetic-quadratic mean and other “means by limits of means”

For $x,y$ positive real numbers, and $p\neq 0$ real, define the Hölder $p$-mean $$M_p(x,y) := \left(\frac{x^p+y^p}{2}\right)^{1/p}$$ whereas $$M_0(x,y) := \sqrt{xy}$$ is the limit of $M_p(x,y)$ when $p\to 0$, i.e., the geometric mean. Furthermore, when $p,q$ are two real numbers, define $L_{p,q}(x,y)$ as the obvious fixed point satisfying the equation $$L_{p,q}(x,y) = L_{p,q}(M_p(x,y),M_q(x,y))$$ —meaning that $L_{p,q}(x,y)$ […]

Prove an algorithm for logarithmic mean $\lim_{n \to \infty} a_n=\lim_{n \to \infty} b_n=\frac{a_0-b_0}{\ln a_0-\ln b_0}$

Take: $$a_0=x,~~~~b_0=y$$ $$a_{n+1}=\frac{a_n+\sqrt{a_nb_n}}{2},~~~~b_{n+1}=\frac{b_n+\sqrt{a_nb_n}}{2}$$ Then we obtain as a limit the logarithmic mean of $x,y$: $$\lim_{n \to \infty} a_n=\lim_{n \to \infty} b_n=\frac{x-y}{\ln x-\ln y}$$ I don’t know how to prove this. But I do know that numerically it fits really well. In fact, the best approximation is obtained if we take geometric mean of $a_n,b_n$: $$x=5,~~~~y=3$$ […]

Mean and variance of the order statistics of a discrete uniform sample without replacement

In answering calculate the mean and variance of the highest number drawn on lottery based on the lowest number drawn, I couldn’t find the mean and variance of the order statistics of a discrete uniform sample without replacement anywhere, so I figured I’d derive them here for future reference. Draw $k$ distinct numbers uniformly from […]

Is it true that $\limsup \phi\le\limsup a_n$, where $\phi=\frac{a_1+…+a_n}n$?

This question already has an answer here: If $\sigma_n=\frac{s_1+s_2+\cdots+s_n}{n}$ then $\operatorname{{lim sup}}\sigma_n \leq \operatorname{lim sup} s_n$ 2 answers

Computing square roots with arithmetic-harmonic mean

We know that if we iterate arithmetic and harmonic means of two numbers, we get their geometric mean. So, basically if we need to compute the square root of $x$: $$\sqrt{x}=\sqrt{1 \cdot x}=AHM(1,x)$$ $$a_0=1,~~~~b_0=x$$ $$a_{n+1}=\frac{a_n+b_n}{2},~~~~~b_{n+1}=\frac{2a_nb_n}{a_n+b_n}=\frac{a_nb_n}{a_{n+1}}$$ As far as I know, this expression will converge for any real positive $x$. See this and this question for […]

Extending the ordered sequence of 'three-number means' beyond AM, GM and HM

I want to create an ordered sequence of various ‘three-number means‘ with as many different elements in it as possible. So far I’ve got $12$ ($8$ unusual ones are highlighted): $$\sqrt{\frac{x^2+y^2+z^2}{3}} \geq \color{blue}{ \frac{\sqrt{x^2+y^2}+\sqrt{y^2+z^2}+\sqrt{z^2+x^2}}{3 \sqrt{2}}} \geq $$ $$\geq \color{blue}{\frac{\sqrt{(x+y)^2+(y+z)^2+(z+x)^2}}{2 \sqrt{3}} } \geq \frac{x+y+z}{3} \geq $$ $$ \geq \color{blue}{ \frac{\sqrt[3]{(x+y)(y+z)(z+x)}}{2} } \geq \color{blue}{\sqrt{\frac{xy+yz+zx}{3}}} \geq $$ $$\geq […]