Questions tagged [expected-value]
The expected value of a random variable is a weighted average of all possible values a random variable can take on, with the weights equal to the probability of taking on that value.
2,344
questions
0
votes
0
answers
46
views
Expected value of iid squared conditioned on sum
I would be interested in finding the value of the following expression:
$$\mathbb{E}[X_k^2\mid S_N]$$
where $X_k$ are iid random variables with $\mathbb{E}[X_k]=\mu$ and $\operatorname{Var}[X_k]=\...
- 91
0
votes
0
answers
23
views
How is summation by parts technique used in this derivation?
In this answer, whuber comments that the technique used in the answer is summation by parts:
The discrete case, assume that $X \ge 0$ takes non-negative integer
values. Then we can write the ...
- 3
0
votes
0
answers
19
views
Expected Value of Continuous Data in R
I am currently working with data involving three continuous variables in R, and I want to calculate the expected value of the joint probability distribution.
I attempted to use the ...
홍민영
0
votes
0
answers
16
views
Second moment of weighted average of random variables
I stumbled upon problem 254 from the SOA Exam P list in
https://www.soa.org/globalassets/assets/Files/Edu/edu-exam-p-sample-quest.pdf
for which I am puzzled by the solution described in
https://www....
- 1
2
votes
1
answer
32
views
analytical asymptotic approximation of the expected maximum, mean, and minimum distance of nearest neighbours in unit ball
Say I uniformly at random distribute $x = n^3$ (independent identically distributed) points in a ball of radius $r=1$ in $\mathbb{R}^3$.
What can be said about the expected maximum, minimum, and mean ...
- 255
1
vote
0
answers
27
views
how can predictive distributions be considered as expectations?
I guess that the prior and posterior predictive distributions can be considered expectation of $p(y|\theta )$ (in case of prior predictive distribution) and $p(\widetilde{y}|\theta )$ (in case of ...
1
vote
1
answer
45
views
Expectation of binomial random variable
Having trouble understanding something I read in a paper recently.
Say we have $X \sim \operatorname{Binomial}(N,p).$ The paper states:
$$E[X \mid N,p] = Np$$ (so far so good)
and
$$E[X] = \mu p$$
...
- 23
1
vote
0
answers
24
views
Convex predictive mean of Gaussian Process
In Gaussian process (GP) regression, predictive mean is
$$ K(X^*,X)[K(X,X)+\sigma^2I]^{-1} \textbf{y}$$
Is there a method to ensure that the predictive mean is convex with respect to the test input $X^...
- 11
1
vote
0
answers
34
views
I don't think this is conditional dependence, so what is it?
I am looking for the name of the following phenomenon. There are three random variables, $X,Y,Z$. We have $P(X,Y) \neq P(X)P(Y)$ and $P(Y,Z) \neq P(Y)P(Z)$. In other words, $X$ and $Y$ are dependent, ...
- 111
0
votes
1
answer
46
views
I would like some insight into what I have been working on here
I work in roofing sales which involves door-knocking at the entry level. Our daily numbers are printed in our GroupMe chat for our branch. I took it upon myself to do some analysis on those numbers. ...
2
votes
1
answer
36
views
Expectation under convex order by multiplying
I am trying to understand if the following statement is true, or the conditions under it is satisfied. Let $M,N$ and $X>0$ be random variables. If the following inequality holds for any concave non-...
- 157
0
votes
0
answers
21
views
Computing a Confidence Interval for E[X] when PMF is given
I am given a Probability Mass Function for a discrete random variable.
From the PMF I computed the Expected Value $E[X]$, the Variance $V[X]$ and the Standard Deviation $S[X]$.
Here is an example (the ...
- 566
2
votes
1
answer
145
views
Expected Value Chi Square distribution
I'm trying to simulate the distribution from the sample variance $s^2$ and compare it with the theoretical distribution.
Therefore, I perform a fairly simple simulation (upfront, I'm not a ...
- 33
2
votes
1
answer
48
views
Expectation under convex order
I am trying to understand if the following statement is true. Let $M,N$ and $X$ be random variables. If the following inequality holds for any concave non-decreasing function $u$
\begin{equation}
\...
- 157
1
vote
1
answer
36
views
Distribution of outcomes of multiple binomial distributions
I have a sample of 50 subjects, where every subject completes a task with two possible outcomes (left or right hand use, with 50% probability) 30 times. On an individual level, this leads to a ...
- 11
0
votes
0
answers
25
views
Expected value of a decreasing function of two random variables
My question is exactly equal to the question posted at Expected value of decreasing function of random variable versus expected value of random variable with just one extra assumption: the two random ...
- 1
0
votes
0
answers
39
views
Bayesian Fubini Tonelli
I am working on a bayesian framework where I place a Gaussian Process on my function $f\sim GP$ and have data $D^n=\{(X_i,Z_i,W_i)\}^n$.
I then have the posterior measure $\mu(f|D^n)$. The posterior ...
- 90
5
votes
1
answer
37
views
Validating binary prediction model
Suppose we have a model that predicts for binary event $e$ ($0$ or $1$) with a single output $p$ (the expected probability $e$ occurs).
If we are able to compare $p$ with the true value of $e$ ($0$ or ...
- 151
1
vote
3
answers
49
views
Deriving home vs away goals - using total expected goals and home/draw/away probabilities
In the context of a football ("soccer") match, if I have the following for a single game:
Probability of Team A winning
Probability of Team B winning
Probability of a draw
The total goals ...
- 13
5
votes
1
answer
104
views
$E[(X+Y)^{a}] > E[(X)^{a}]$?
Assume I have two strictly positive i.i.d. random variables, $X$ and $Y$. Under what conditions is the following inequality true?
$$E[(X+Y)^{a}] > E[(X)^{a}], \hspace{2mm} a \in (0,1)$$
Should have ...
- 263
3
votes
1
answer
275
views
Probability algorithm on strings
Let $x$ be any binary string $\in (0,1)^*.$
The majority language is given by:
$$\text{MAJ}:=\{x\in (0,1)^*:\sum_{i=1}^ {|x|}x_i>\frac{|x|}{2}\},\text{where $x_i$ is the $i$-th position value(...
- 69
5
votes
1
answer
114
views
Need help in calculating $\mathbb{E}(\frac{1}{x_{(2)}-x_{(1)}}\int_{x_{(1)}}^{x_{(2)}} f(t) \ dt)$, where $x_{(i)}$ are related Beta distribution
Suppose $Y, Z \stackrel{\text{iid}}{\sim}\mathrm{Uniform}(0,1)$.
Let $a = g(\min(y,z)),\ b=g(\max(y,z)).$
How can I calculate the expectation $$\mathbb{E}\left[\frac{1}{b-a}\int_a^b f(t) \ dt\right]$$ ...
- 171
0
votes
0
answers
78
views
Derive the expectation and variance of squared sample correlation: delta-method or else?
I would like to obtain the expectation and variance of the squared Pearson sample correlation ($\operatorname{E}(R_{lk}^2)$ and $V(R_{lk}^2)$) between two random variables $l$ and $k$ following a ...
- 215
1
vote
1
answer
26
views
Why does $E(V_n/(n+2)-1)^2=2/(n+2)$ when $V_n\sim\chi^2(n)$?
I was reading some lecture notes when I saw a simplification I didn't understand. Specifically, we have $V_n\sim\chi^2(n)$. It was then written then
$$E\left(\frac{1}{n+2}V_n-1\right)^2=\frac{2}{n+2}.$...
- 155
0
votes
0
answers
47
views
Expectation of $u^\top(u+Ax)$, when $A$ and $u$ are nonlinear functions of $x$
Let $x\in\mathbb R^d$, and $s=\operatorname{softmax}(x)$. Let $y$ be a fixed one-hot vector such that
$$u = s-y \\
v =(\operatorname{diag}(s) - ss^\top)x$$
I am interested in the inequality $u^\top (u ...
- 101
1
vote
0
answers
16
views
Expected value of Cosinus in High dimension
I would like to prove that the cosinus of the angle formed by 3 randomly points tends to $\frac{1}{2}$ as the dimensionality tends to $\infty$. Could it be solved with the expected value formula ? It ...
- 11
0
votes
0
answers
12
views
Expectation over cost-normalized Expected improvements
Are the following two expressions equivalent if we assume the independence of f(x) and C(x)?
$$
E\left[\frac{E\left[\max\left(f(x) - f(x^*), 0\right)\right]} {C(x)}\right]
$$
$$
\frac{E\left[\max\...
1
vote
0
answers
33
views
Expectation & Covariance matrix of indicator vector
Suppose we have the $p$-dimensional random vector $\boldsymbol{X} \sim \mathcal{N}(\boldsymbol{\mu}, \Sigma)$. Take the set $A$ to be (without loss of generality) the negative real line, thus $A = (- \...
- 55
1
vote
0
answers
68
views
How to calculate the expectation of the following Dirichlet distribution and Beta distribution?
This is a question from my research, related to the derivation of the variational EM algorithm with mean-field assumption about LDA-based model.
We all know, given that $\boldsymbol{\theta} \sim \...
- 11
0
votes
0
answers
21
views
Verifying the integrability condition of a deterministic volatility function
Suppose there is integrability condition:
\begin{equation}
\mathbb{E}\left[\int_0^T\frac{\sigma^2(t)}{T-t}dt\right]<\infty
\end{equation}
for an arbitrary volatility function. Suppose I nominate ...
- 1,226