18 votes
Accepted

Suppose I have 100 integers and I sample 10 without repetition. What is the expected rank of the lowest out of 10 samples?

To obtain an answer we must know how many ties there are among these $100$ integers and where they occur: that's too complicated and likely is not the intent of the question. (Nevertheless, the ...
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  • 287k
8 votes
Accepted

Expectation of the ratio of sum (XY) and sum(X)

I will assume $a=0$ and $b=1$ in the following. Here is a simulation experiment to look at the variability of the expectation in $M$: ...
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6 votes

Expectation of the ratio of sum (XY) and sum(X)

In your case of $\text{sum}(XY)/\text{sum}(X)$ you have that the $X$ and $Y$ are correlated. We can rewrite it in a different form such that we have a similar weighted average expression but with ...
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2 votes

In a time series $x_t, x_{t-1},...,$, why is $E[x_t|x_t, x_{t-1},...]= x_t$?

As the comment by J. Delaney says, the result would hold even without conditioning on $x_{t-1}, x_{t-2},\cdots$. The statement in the OP's question might be the result of a bad cut-and-paste job from ...
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2 votes
Accepted

what is the expectation value of a norm of a random variable from the standard normal distribution?

Since the components are independent, it means that the squared $L_2$ norm of $\zeta$ has a $\chi^2_k$ distribution, which is the sum of squared independent standard normal variables. The expected ...
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  • 30.9k
2 votes
Accepted

Bayesian Quadrature to find expectation of unkown function w.r.t. known pdf

This type of problem can often be addressed by a transformation of variables. For this you need to find an injective almost everywhere differentiable map $\Phi:\mathbb{R}^d\rightarrow ]0,1[^d$ with $\...
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2 votes
Accepted

Interpretation problems of linear model with no predictors

The linear regression model is $$ Y_i = \beta_0 + \beta_1 X_1 + \dots + \beta_k X_k + \varepsilon_i $$ with $\varepsilon_i$ being i.i.d. Gaussian noise with mean equal to zero. The model estimates the ...
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1 vote

Expectation of a random variable with positive density at infinity?

This is a case where you are dealing with a non-negative random variable on the extended real numbers.$^\dagger$ In this case the expected value is: $$\mathbb{E}(X) = \int \limits_0^\infty x f(x) dx +...
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