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Questions tagged [mean]

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2
votes
1answer
63 views

Independence of r.v.'s following a distribution that is the ratio between complex Gaussian and Chi-square r.v.'s

Given the following two R.V.s $$z_1 = \frac{x_1}{|x_1|^2 + |x_2|^2 + \cdots + |x_M|^2}$$ and $$z_2 = \frac{x_2}{|x_1|^2 + |x_2|^2 + \cdots + |x_M|^2}$$ where $x_i \sim \mathcal{CN}(0,a), \forall i$...
1
vote
1answer
63 views

Correlation between r.v.'s following a distribution that is the ration between complex Gaussian and Chi-square r.v.'s

Given the following two R.V.s $$z_{1} = \frac{x_{1}}{|x_{1}|^2 + |x_{2}|^2 + ... + |x_{M}|^2}$$ and $$z_{2} = \frac{x_{2}}{|x_{1}|^2 + |x_{2}|^2 + ... + |x_{M}|^2}$$ where $x_{i} \sim \mathcal{CN}(...
1
vote
0answers
27 views

Mean and correlation of product of two random processes

I have two random process: $$A(at)$$ $$\cos(2\pi f_0t+\Phi)$$ with these hypothesis: $a$ and $f_0$ are constant $\Phi$ is uniformly distributed in $[0,\pi)$ $A(at)$ is WSS I must calculate the ...
0
votes
1answer
144 views

The Euler's totient function and the product of distinct primes dividing $n$ versus the Heronian means

For integers $n\geq 1$ with $$\operatorname{rad}(n)=\prod_{\substack{p\mid n\\p\text{ prime}}}p$$ we denote the squarefree kernel or radical of an integer $n$ (see if you want this Wikipedia). And $\...
3
votes
1answer
95 views

Geometric mean of positive measures

Let me given with an obvious example. Let $\Omega\subset{\mathbb R}^n$ be an open domain. If $f,g\in L^1(\Omega)$ and $f,g\ge0$, then $\sqrt{fg}\,\in L^1(\Omega)$. Now let me replace the absolutely ...