6
votes
Accepted
Sign of Correlation between $X$ and $f(X)$ for strictly monotonic $f$
Let $f$ be strictly increasing. Then
$$\operatorname{Cov}(X, f(X)) =\mathbb E[Xf(X) ]-\mathbb E[X]\mathbb E[f(X) ]=\mathbb E[(X-\mathbb E[X])(f(X) -f(\mathbb E[X]))].\tag 1\label 1$$
Now $$X\gtreqless\...
- 5,137
4
votes
Accepted
Sign of Correlation between $X$ and $\log X$
Your sign requirement does not necessarily hold, but it's still possible to prove the result using an alternative method. Since $x \log x$ is convex and $\log x$ is concave (over the stipulated range)...
- 109k
3
votes
Accepted
Variance of the difference of two iid sample means
The sample average $\bar X$ has expected value $\mu_1$ and variance $\sigma^2_1/n$. By the same token, $\bar Y$ has expected value $\mu_2$ and variance $\sigma^2_2/n$. Now using the linearity of ...
- 8,101
2
votes
Accepted
Suppose $Y \sim Ber(X)$ where $X = F(Z)$ and $Z \sim N(\mu, \sigma^2)$. What is the expected value $E[Y-X]$?
What you have done is correct (assuming that $F$ is fixed), but it can be simplified, generalised, and made more explicit. Let's start by generalising your model and writing it in a more explicit ...
- 109k
2
votes
Convergence of moment of functional of random variable
This is not true. Set the probability space $(\Omega, \mathscr{F}, P)$ to be $((0, 1], \mathscr{B}, \lambda)$ and define $X_n = \sqrt{n}I_{(0, n^{-1})}(x)$, $n = 1, 2, \ldots$, $X \equiv 0$, then
\...
- 10.3k
2
votes
Inverse moment of Multivariate Normal Norm
For the case where $\Sigma = \mathbf{I}\sigma$, we have the following formulas:
\begin{equation}
\mathbb{E}\left( \frac{1}{||x||} \right) = \frac{1}{\sqrt{2}} {}_1F_1 \left(\frac{1}{2}, \frac{P}{2}, -\...
- 168
1
vote
Question on solving OLS
Edit: The example appears to be wrong
I am using the setup described in 3.4 and 3.5.1 of Domain adaptation under structural causal models which matches your description.
In the Source environment we ...
- 380
1
vote
Accepted
How do we define the pdf in the multi-variate case and compute expectations?
Short answer: there is no inconsistency. I want to make the following two remarks:
Merely from the differentiation perspective, your confusion is understandable. For example, consider a bivariate ...
- 10.3k
1
vote
Accepted
Expected value of Y = (1/X) where X is Gamma Distribution
This is straightforward integration:
Note that since $\mathbb{E}(X) = \alpha\beta$ then the $\beta$ is a scale parameter. Thus
$$
\begin{aligned}
M_X(t) =& (1-\beta t)^{-\alpha}\\
\therefore\...
- 436
1
vote
Accepted
Evaluating $E(x^{-1}). $
$$E\left(\frac{1}{x}\right)=\int_{-\infty}^{0}{e^{ux}du}=\int_{0}^{\infty}{e^{-ux}du}$$
$$E\left(\frac{1}{x}\right)=\int_{0}^{\infty}{x^{-1}f\left(x\right)dx=\int_{0}^{\infty}{\left(\int_{0}^{\infty}{...
- 31
1
vote
What is the expectation of $e^X$, where $X$ is a random variable with a geometric distribution?
Building on the answer by @Stephan and comment by user whuber. First, there is two versions of the geometric distribution; for now I use the one with support $0,1,2, \ldots$ which has moment ...
- 71k
1
vote
Expected squared distance between order statistics?
The terms of this sum are known exactly for some distributions, since
$$E[(X_{(i)}-Y_{(i)})^2]=E[X_{(i)}^2]-2E[X_{(i)}]E[Y_{(i)}]+E[Y_{(i)}^2]=2\text{Var}[X_{(i)}].$$
For those distributions, we can ...
- 2,227
1
vote
Does the unconditional mean of a non stationary ARMA process exist?
You are right, the random walk with no drift has mean zero, or the starting value if such a value is given. As you said, the random walk is just the sum of i.i.d. random variables $\epsilon_t$ with ...
- 268
Only top scored, non community-wiki answers of a minimum length are eligible