# 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.

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### $E[X^T (Y-Z)] = E[X^T] E[Y-Z]$ but what about $E[(X^T (Y-Z))^2]$?

Let $X, Y$, and $Z$ be random vectors with $X$ independent of $Y$ and $Z$. Due to the independence we have $$E[X^T (Y-Z)] = E[X^T] E[Y-Z].$$ But what what $E[(X^T (Y-Z))^2]$? Is it possible to ...
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### Expected value of a conditional normal [closed]

Why $E[(Y- E[Y|Z])^2] = \sigma^2_Y(1-\rho^2)$????? I know this is the variance of the conditional distribution $Y|Z$. I thought $E[(Y|Z- E[Y|Z])^2] = \sigma^2_Y(1-\rho^2)$, but the slides says ...
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### Expected value of a bivariate distribution as an integral

Let's assume an absolutely continuous random variable, $X$, with PDF $f(x)$. $$\mathbb{E}\big[X\big] = \int_{\mathbb{R}}xf(x)dx$$ If $X\sim f(x_1,x_2)$ is multivariate, then it makes sense to me to ...
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### Obtaining expectation of a variable given a conditional expectation

Is there any mathematical association between the conditional expectation of a variable given another variable, and the unconditional expectation of that variable? I realise that given a joint ...
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### Expected value (Mean) of a joint distribution

I saw in a textbook that if we have a joint distribution $f(X,Y)$ that is a Gaussian distribution, then we have the mode equal to the mean. The mode is just the values of $X$ and $Y$ such that $f(X,Y)$...
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### Problem calculating expectation using law of total expectation

I'm confusing myself with conditional expectation and could really use your help! I am trying to calculate an expectation that arises in the context of doing variational inference. However, the ...
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### Independence of variables between X and Y^X

If X and Y are independent, then are X and Y^X independent? Does the realisation of X have to be the same as the X in the power of Y? I think this question sounds silly but I'm trying to clear a major ...
Background. Let $V = (X,Y)$ be a random vector in 2-dimensions uniformly distributed over two disjoint regions $R_X \cup R_Y$ defined as follows:  \begin{align} R_X &= ([0,1] \times [0,1]) \...