All Questions
Tagged with expected-value mathematical-statistics
217
questions
4
votes
2
answers
134
views
What conditions are there on the exponent $p$ such that $\underset{\mu}{\arg\min}\left\{\mathbb E\left\vert X-\mu\right\vert^p\right\} $ must exist?
Let $X\sim F(x)$ be a (univariate) random variable defined by distribution function $F$. If the expected value exists, it is equal to $
\mathbb E[X] = \underset{\mu}{\arg\min}\left\{\mathbb E\left\...
- 65k
2
votes
3
answers
105
views
Scaling the conditioned random variable does not change conditional distribution, why?
Given two random variables $X$ and $Y$, I know intuitively that
$$
\mathbb{E}[X\,|\,Y]=\mathbb{E}[X\,|\,cY],
$$
where $c$ is some non-random constant. My intuition tells me that scaling the ...
- 229
2
votes
1
answer
60
views
Expectation and variance of bivariate skew normal distribution
I am fitting a bivariate skew normal distribution to a 2D data through the sn package in R. I get a $2 \times 1$ vector of ...
- 163
0
votes
0
answers
23
views
Question regarding probability and maximum possible variance
I have the following question:
Suppose we have a set of 10 numbers (1, 2, ... , 10), each with a certain probability tagged to it.
Is it true that the highest possible variance is achieved when 1 and ...
- 31
1
vote
0
answers
20
views
Is There a Standard Metric for Evaluating Treatment Impact Considering Action Cost in Uplift Models?
I'm currently exploring Uplift modeling, specifically the use of the Conditional Average Treatment Effect (CATE) metric:
$$ \tau(t', t, x) := \mathbb{E}[Y | X=x, T=t'] - \mathbb{E}[Y | X=x, T=t] $$
...
- 57
1
vote
1
answer
164
views
Show that for random variable $X$ with $N = \{1, 2, \ldots \}$, $E(X) = \sum_{n = 1}^\infty P(X \geq n)$ [duplicate]
Prove that for random variable with natural numbers from 1 to infinity the expected value $E(X)$ is equal to $\sum_{n = 1}^\infty P(X \geq n)$. Is this the mathematically correct way to prove it? And ...
- 45
0
votes
1
answer
54
views
Calculate $E[X]^2$ where $X \sim \operatorname{Binomial}(n,p)$ with binomial coefficients expansion [closed]
Calculation of $EX$ using the binomial expansion formula is easy:
\begin{align}
EX &= \sum_{x=0}^{n}x\frac{n!}{(n-x)!x!}p^{x}(1-p)^{n-x}\\& = np \sum_{x=1}^{n}\frac{(n-1)!}{(n-x)!(x-1)!}p^{x-...
- 11
0
votes
1
answer
81
views
How to tell if an estimator is unbiased? How to find expected value of an estimator?
You come up with a great idea of an estimator for $\beta_1$ in the SLR model which satisfies SLR.1 to SLR.4:$$y_i=\beta_0+\beta_1x_i+u_i$$ Given a sample $\left\{(x_i,y_i),i=1,2,3,\dots,n\right\}$, ...
7
votes
1
answer
208
views
Expected absolute deviation greater than standard Laplace
Could there exist a distribution, other than standard Laplace (probability density of the form $1/2e^{-|x|}$), on $\mathbb{R}$ such that $E[x]=0,E[|x|]=1$ and that
\begin{equation*}
E[|x-a|] \geq |a|+...
- 155
0
votes
0
answers
26
views
Showing conditional expectation equivalence
Question: I have a random one-hot vector $B\in\{0,1\}^L$, where the position of the one entry follows $\text{Categorical}(\pi)$ where $\pi\in\mathbb{R}^L$. Given a constant (non-random) $L\times L$ ...
- 229
2
votes
1
answer
135
views
Showing incompleteness of density
You observe a sample of 100 independent observations $X_i$ from a population with the density
$$
g(x)=C \sqrt{\lambda} \exp \left(-\lambda x^2-\lambda^2 x^4\right), \quad-\infty<x<\infty
$$
...
- 133
2
votes
1
answer
70
views
Certain approximation in the setting of three expectation values does not make sense to me
I'm currently going through some lecture notes in the field of Bayes optimization and I'm currently looking at a expression looking like this:
$$\mathbb{E}_{x^*} \left[\mathbb{E}_y\left[\left\{\mathbb{...
1
vote
1
answer
94
views
van der Vaart Asymptotic Statistics, page 38, why does $e_\theta'=\operatorname{Cov}_{\theta}t(X)$?
On Page 38 of van der Vaart's Asymptotic Statistics (near the bottom of the page), it says
By differentiating $E_\theta t(X)$ under the expectation sign (which is justified by the lemma), we see that ...
- 2,966
1
vote
1
answer
52
views
Equivalence of expectations
I have two independent random variables $X$ and $Y$, and a constant term $a$. Furthermore, $\mathbb{E}[Y]=0$. I want to show that
$$
\mathbb{E}\Big[\mathbb{E}[X+Y/a|aX+Y]\cdot\mathbb{E}[X|aX+Y]\Big]
=\...
- 229
1
vote
1
answer
64
views
Upper bound for $E[X^2 1_{|X| > v }]$
Given a random variable $X$ such that for some $p$, we have
$$
|| X ||_{L_p} = (E |X|^p )^{1/p} \le C_p < \infty.
$$
Supposedly then the following statement is true: for any $v>0$
$$
E (X^2 1_{|...
- 1,501