15
votes
Is the expectation of a random vector multiplied by its transpose equal to the product of the expectation of the vector and that of the transpose
To confirm a claim is not true, you don't "prove it". Instead, just provide a counterexample would be sufficient.
You are actually on the right track. Any random vector $z$ with positive ...
- 16.8k
5
votes
Is the expectation of a random vector multiplied by its transpose equal to the product of the expectation of the vector and that of the transpose
Here's a discrete counter-example, if that's easier to wrap your head around.
Let
$$
\begin{align}
P(z_0=1 \wedge z_1=1) &= 0.5 \\
P(z_0=-1 \wedge z_1=-1) &= 0.5
\end{align}
$$
Then
$$
E[z] = ...
- 151
2
votes
Covariance of a linear and quadratic form of a multivariate normal
The covariance between a linear and a quadratic form of a multivariate normal vector is given in Mathai and Provost, page 74, Theorem 3.2d.2.
Let $\mathbf{y} \in \mathbb{R}^p \sim \mathcal{N}(\mathbf{\...
- 1,168
1
vote
Calculate the variance of $\sum\limits_{i=1}^{n-1} \sum\limits_{j=i+1}^n S(X_i - X_j)$ for $X_1,\ldots,X_n$ i.i.d. random variables
Since the function $S$ is symmetric about $0$, if we denote $S(x - y)$ by $\phi(x, y)$, then the $U$ in your question can be viewed as a (scaled) U-statistic with the kernel $\phi$:
\begin{align*}
U = ...
- 16.8k
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