# Questions tagged [conditional-expectation]

A conditional expectation is the expectation of a random variable, given information on another variable or variables (mostly, by specifying their value).

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### Proof that when the conditional mean of Y does not depend on X, Y and X are uncorrelated

I've been struggling to understand the proof that when E(Y|X) = $µ_y$, then cov(Y,X) = corr(Y,X) = 0, as provided in Stock and Watson's Introduction to Econometrics, 4th Edition. The book says the ...
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### Expectation of a random variable over disjoint intervals

I wanted to know if there is a formal way to show the following. Say I have a random variable X that takes value over the interval ...
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### Show that the manhattan distance from the origin of an ubiased random walk in $\mathbb{Z}^2$ defines a martingale sequence

Consider the infinite lattice $N \times N$. A pebble starting at the origin walks at random, each time moving equiprobably to one of its four neighbors. Let $X_i$ be the distance from the origin, ...
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### Why isn't every nonparametric model with random model design an additive noise model?

Let $Y$ be a real random variable and $X$ be a real random vector. In a nonparametric model with additive noise, we assume the relationship $$Y = f(X) + \epsilon$$ for some unknown regression function ...
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### Interpretation of coefficients in multivariate Poisson regression in terms of expectation

If I have a regression model $Y = \beta_0 + \beta_1 A + \beta_2 B$, where $A$ and $B$ are treatment indicators. Firstly, if I frame the definition of $\beta_1$ in terms of expectations, how do you ...
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### Simplifying the conditional expectation $E[X|Y]=(-a +Y-E[e|Y])/b$ to find the slope of the Reverse Regression line

I have a pretty basic question about conditional expectation that is stumping me. Consider the real-valued random variables $Y$, $X$ and $e$, where $E[e] = 0$ and $X$ and $e$ are independent. Assuming ...
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### Conditional expectation of two regions

In $\mathbb{R}^2$, define square region A with corners at ${(0.5,0),(1.5,0),(1.5,1),(0.5,1)}$ and and square region B with corners at ${(0,1), (1,1),(1,2), (0,2)}$. Suppose that random variables X and ...
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### Projection operator: why squared norm of the sum of them is equal (or smaller) than the sum of the squared norms?

I am working through the proof of Lemma 2 in this paper (page 25, need it for my own research) and I am stuck at the very first step. Here, I will formulate a bit simplified version of this step. ...
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### Conditional Expectation : How is E[E[xy|x]]=E[xE[y|x]]?

I read that "The law of total expectations states E[xy]=E[E[xy|x]]. By linearity of conditional expectations, E[E[xy|x]]=E[xE[y|x]]" but I am not able to understand the part "E[E[xy|x]]=...
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### What is an example of a loss function that is not minimized by the conditional expectation?

From statistical decision theory we know that if we want to minimize EPE (Expected prediction error) it is sufficient to minimize the conditional expectation of the loss function. \$f(x) = argmin_{c} ...