2 votes

$\mathbb{E}[X^2]\leq k \mathbb{E}[X]^2$, upper bound second moment from first moment

It's easy to show that this holds for Bernoulli random variables. By the contrary, the family of Beroulli random variables shows that a fixed finite $k$ cannot exist. The argument goes the same as ...
Zhanxiong's user avatar
  • 18.9k
1 vote

Applying the law of total variance

The first term is the variance caused by $Y$ varying, the second term is the variance caused by $X$ varying. To calculate the second term, find the expectation of $Y$ for each value of $X$, and then ...
George Savva's user avatar
  • 2,400
1 vote

Variance partitioning with crossed and nested factors: A simulation study

I've come across another option to simulate data from a linear mixed-effects model: designr::simLMM. (Aside: One weakness of the {designr} package is that Google is ...
dipetkov's user avatar
  • 9,910
1 vote

Convergence in distribution versus convergence of moments

For a counterexample let $(X_n)_{n \in \mathbb N_{\geq 1}} \mathrel{:=} (n \cdot \mathbf 1_{[0, n^{-2}]})_{n \in \mathbb N_{\geq 1}}$ be a sequence of random variables each defined on the probability ...
statmerkur's user avatar
  • 5,970

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