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Difference between $\mathbb{E}[Y|X]$ and $\mathbb{E}[Y|X=x]$

𝔼[𝑌|𝑋] is a random variable of X, where 𝔼[𝑌|𝑋=𝑥] is a constant when the random variable X is evaluated at x of its domain.
Colin Fang's user avatar
6 votes
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Difference between $\mathbb{E}[Y|X]$ and $\mathbb{E}[Y|X=x]$

In measure-theoretic probability, it is important to understand that while $E[Y|X]$ is a carefully defined mathematical object (as a Radon-Nikodym derivative), "$E[Y|X = x]$" is merely a ...
Zhanxiong's user avatar
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3 votes

Law of the unconscious statistician for conditional expectation and pushforward measure of conditional distribution

Before answering your formal questions, it is extremely important to clarify the (difficult!) notion of conditional distribution first. In order to do so, let me quote its rigorous definition from ...
Zhanxiong's user avatar
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3 votes

Law of the unconscious statistician for conditional expectation and pushforward measure of conditional distribution

Let me provide you a succinct glimpse of the utility of regular conditional distributions. (And for sanity, I am not involving $Z$ here; also unless specified, notations used would be common and hence ...
User1865345's user avatar
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-3 votes

Law of the unconscious statistician for conditional expectation and pushforward measure of conditional distribution

$$\mathbb{E}[g(X)|Z]=\int_{\Omega}g(X)dP_{\Omega/Z}=\int_{\mathcal{X}}g(x)dP_{X/Z}$$ $$\mathbb{E}[g(X)|Z=z]=\int_{\Omega}g(X)dP_{\Omega/Z}(z)=\int_{\mathcal{X}}g(x)dP_{X/Z}(z)$$
Speltzu's user avatar
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0 votes

Probability extraction from random forest classifier

What you reference is an assessment of output calibration and calibrating outputs that are not calibrated. By calibration, it is meant that a predicted value $\hat p_i\in[0,1]$ truly corresponds to an ...
Dave's user avatar
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1 vote
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How to derive conditional distribution function on range from conditional distribution function on a single value

Suppose the joint pdf of $(X, Y)$ is given by $f(x, y)$, and $P(y_1 < Y < y_2) > 0$, I am assuming you are interested in determining the conditional distribution of $X$ given the event $B: = \...
Zhanxiong's user avatar
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1 vote

Conditional probability, who's right?

A quick simulation of 10M bits in R to confirm the answers given so far: ...
jblood94's user avatar
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1 vote

Conditional probability, who's right?

Executive Summary I believe answer 2 is correct in that a) is 0.9 and b) is 0.8. Setup Let $A$ be the probability that the sent bit was 0. Thus $\bar{A}$ is the probability that the sent bit was 1. ...
Avraham's user avatar
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0 votes

Conditional probability, who's right?

What if I said that answer 2) is correct? Let's look at the known probabilities: $P(0\,\mathrm{received}|0\,\mathrm{sent})=9/10$ $P(1 received|0 sent)=1/10$ $P(1 received|1 sent)=4/5=8/10$ $P(0 ...
jginestet's user avatar
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1 vote
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The Math Behind the Conditional Probability of a Probabilistic PCA

The model factorizes over observations. In other words, this probabilistic representation is only about one multi-dimensional sample: Thus $x$ is $D \times 1$, $W$ is $D \times M$ and $z$ is $M \times ...
Firebug's user avatar
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1 vote

Conditional Expectation of Product of Normals given a Linear Combination

Given a linear combination like $\xi-2\eta =Y$ (say) of independent standard normal variables $(\xi,\eta)$, you can always find another linear combination $X$ of $(\xi,\eta)$ such that $X$ and $Y$ are ...
StubbornAtom's user avatar
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1 vote

conditional probabilities in chess

I have concerns about this, but a simple approach is to regress the win/loss outcomes against the types of games, such as with a logistic regression. This then turns the problem into what might be a ...
Dave's user avatar
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1 vote

Distribution of a random variable conditional on its being a maximum or not

This is a semi-informal answer for Q2. Consider the case of $\epsilon_1,\cdots,\epsilon_D\overset{iid}{\sim}\text{Gumbel}(\mu,\beta)$, with $D\geq 2$. Then, as $\epsilon_j$ are absolutely continuous, $...
Johan de Aguas's user avatar
0 votes

Distribution of a random variable conditional on its being a maximum or not

The condition $Y=h$ does not truncate any $\epsilon_j$: any value for $\epsilon_j$ can still be exceeded by another variable. The condition $Y=h$ leads to a simple formula for any iid $\epsilon$’s. ...
Matt F.'s user avatar
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5 votes

How to obtain $p(x)$ given samples from $p(y|x)$ and $p(y)$?

Depending on what we mean by specifying the conditional distribution, one possible answer is that we can't in general. If we specify the conditional distribution for $Y$, without specifying what the $...
Glen_b's user avatar
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About the use of Bayes' rule for continuous valued random variables

Bayes rule is stated for events, and the event you condition on is usually taken to have non-zero probability. e.g. https://en.wikipedia.org/wiki/Bayes%27_theorem $$ P\left[A\,\big|\,B\right]=\frac{P[...
Cryo's user avatar
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13 votes
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How to obtain $p(x)$ given samples from $p(y|x)$ and $p(y)$?

This problem is equivalent to solving a Fredholm integral equation of the first kind. This is, solving for $p(x)$ such that: $$ p(y) = \int_{\text{supp}(X)} p(y\mid x)\, p(x)\, \text{d}x,\quad \forall ...
Johan de Aguas's user avatar
0 votes

Deriving the conditional distributions of a multivariate normal distribution

Since I regularly come back to this, here is my attempt to reduce the gore of Ben's answer (wether or not you think this is true will depend on your tastes). With this approach it turns out to be a ...
Felix B.'s user avatar
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