# Tagged Questions

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### Properties of logarithmic mean.

I have been studying the logarithmic mean for the last few days now. Could someone please help me with the following two questions? 1) We know that the log mean is in between the geometric mean and ...
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### Distribution of higher powers than 2 of a gaussian distribution

If $X \sim \mathcal{N}(0,1)$, then $X^2 \sim \chi^2(1)$. What about higher powers of $X$? I know that the Gamma Distribution is a generalization of the $\chi^2$ distribution, but I don't know how the ...
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### How to calculate the summation of this sequence

$$\sum_{j=0}^n {n \choose j}e^{iuj}p^j(1-p)^{n-j}$$ Here, $i$ stand for complex number $i$, $j \in N$, and $0<p<1$. Since it is a sequence, I coundn't find any formula for this. The answer is ...
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### Probability computation, tossing two dice

I have some ideas on how to solve the problem, but simulations do not support my analytical results :) Toss two dice and sum their value and write it down: Denote by $X_n$ the result at $n$-th toss. ...
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### How to calculate that series

I was looking at the solution of a problem, then this: I don't know how to caluculate that series in the denominator, and here I assume the result is done by write out that series. Here X is a ...
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### When does equality in Markov's inequality occur?

Markov's inequality states that given any nonnegative random variable and $a>0$ then we have: $$P(X \geq a) \leq \frac{E(X)}{a}$$ At which $a$ is equality supposed to hold?
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### Convergence condition of infinite cosine product

Please show that, given that $\sum_{k\ge1}c_k^2=\infty$ and $c_k\rightarrow 0$, $$\lim_{n\rightarrow\infty}\prod_{k=1}^n\cos{tc_k}=0$$ for every $t\neq0$. (All variables here are real numbers.) The ...
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### Almost everywhere convergence of a function of rv

Let f(x) a continuous function on $\mathbb {\bar{R}^+}$(extended positive real line, $x\in(0,\infty]$). Take $y\in \mathbb R^+$. We can say that $\lim_{y\to 0}\frac{x}{y}=\infty$ almost everywhere ...
Given a random variable $X$, if we take a measurable and bounded function $f(X)$ then can we say that $f$ is Lebesgue integrable wrt a probability measure on $\mathbb R$? In Real Analyses book by ...