All Questions
Tagged with expected-value conditional-expectation
198
questions
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45
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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
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0
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18
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Expected average distance in greedy matching on a circle
Now we have several independent and identically distributed random variables following the uniform distribution on the interval [0, 1].They are denoted as $x_1, x_2, x_3, ..., x_m$ and $y_1, y_2, ..., ...
2
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3
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105
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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
1
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2
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95
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In this RL problem, why is the substitution $q_*(A_t)=\mathbb{E}[R_t | A_t] \to R_t $ valid within this expectation (over actions)?
The question that follows is from a machine learning textbook (Reinforcement learning Suttion and Barto page 39 link).
Given:
a probability distribution over actions $x$ (a policy) at time $t$ ...
- 1,617
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62
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Confused on Kullback-Leibler divergence being invoked without proper definition
I am trying to understand how authors of the DDPM paper in appendix A, made the leap from equation 21 to equation 22.
Specifically, it is not clear to me how they managed to convert the first term of ...
- 1,805
6
votes
2
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176
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Inequalities involving expectations
Consider four random variables $W, X,Q,Y$, where $Q$ and $Y$ are binary. Assume
$$
\begin{aligned}
& (1) \quad E(Q(X+W)|Y=1)\geq 0\\
& (2) \quad E(W|Y=1)=0\\
& (3) \quad \Pr(Q=1|Y=1,W)=1
\...
- 889
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What is the expected length of an interval on an arc of a circle that can be constructed using exponential variates?
I had asked this question on Math stackexchange once before and now again but this does not seem to be drawing too much attention. Since this is a question that can be safely classified as non-measure ...
- 71
1
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1
answer
40
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Calibration Expectation Decompostion
I am reading a "Calibrated Structured Prediction by Kuleshov and Liang" link.
Calibration and sharpness. Given a forecaster $F : X → [0, 1]$, define $T(x) = \mathbb{E}[y| F(x)]$ to be the ...
- 11
2
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2
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166
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How should I interpret the assumption of the regression?
I read an econometrics book which states one of the basic assumptions of regression is that
$$E(u|x) = 0$$
In another book however I see it written that
$$E(u_i|x_i) = 0$$
Are these two saying the ...
2
votes
1
answer
58
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(Using conditional expectation to calculate) expected value of the product of two dependent random variables
Let $\mathbf{X}$ be Binomial point process in $W = [0, 6] \times [0, 4]$ with $n$ points. Let $A_1 = [0, 2] \times [0, 4]$, $A_2 = [0, 6] \times [0, 2]$, and $A_3 = [2, 6] × [2, 4]$. I want to find $E[...
- 2,096
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26
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Showing conditional expectation equivalence
Question: I have a random one-hot vector $B\in\{0,1\}^L$, where the position of the one entry follows $\text{Categorical}(\pi)$ where $\pi\in\mathbb{R}^L$. Given a constant (non-random) $L\times L$ ...
- 229
2
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34
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Approximating an expectation of a log-product and another expectation
I am dealing with a variation of a standard problem. Given an objective function $O := E_{a \sim P_\theta(A)} [f(a)]$, we can calculate its gradient $\nabla_\theta O$ as follows:
$$\nabla_\theta O = ...
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32
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Help with iterated expectations when deriving diff-in-diff estimand
[Closed, see comments]
I was reading through this paper on recent advances in diff-in-diff and got stuck with a (probably very simple) issue when trying to derive the final population estimand.
In ...
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1
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1
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52
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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
3
votes
1
answer
87
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Difficulties with the PGF of X+Y with Y~Poisson(1) and X~Poisson(Y)
The pair of random variables $(X, Y )$ is distributed as follows. $Y$ has probability mass function $\text{Poisson}(1).$ Given $Y , X$ has probability mass function $\text{Poisson}(Y ).$ Show that the ...
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