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54m
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Mean of probability distribution Incidentally, $m$ and $n$ only seem to appear as their product $mn$ in your expression. |
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56m
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Mean of probability distribution Stirling numbers of the second kind appear in occupancy problems like the coupon collectors problem and the birthday problem, which often have simple expressions for expectations by using linearity of expectation rather than sums like yours (presumably of a discrete distribution on the integers rather than a continuous density). What does your distribution represent? |
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1h
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How to estimate precision and recall without taking a huge random sample when the positive class is relatively rare If you trust the $0.5\%$ figure to be accurate, perhaps from a different study, then you can use that. But if it is only a rough order of magnitude, then large samples may be the way to go. |
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3h
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awarded | Nice Answer |
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14h
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I need a function for my shader It depends on what shape you want in the middle, but you could try something like 1 - cos(x * 3.14) * cos(y * 3.14) or 1 - min(cos(x * 3.14) , cos(y * 3.14))
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14h
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reviewed | Approve suggested edit on I need a function for my shader |
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14h
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revised |
Why there is two different answers for the volume of a frustum? added 26 characters in body |
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14h
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revised |
Why there is two different answers for the volume of a frustum? added 26 characters in body |
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14h
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answered | Why there is two different answers for the volume of a frustum? |
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1d
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Sampling distribution of the proportion of events For binary variables, I would have thought the proportion of one of the categories had a binomial distribution scaled by $\frac1n$ rather a normal distribution. So with more categories, I would have thought you had a multinomial distribution scaled again scaled by $\frac1n$. |
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1d
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Expectation of a non negative integer value random variable @Kolakoski54 The limits of the sum matter: $\mathbb E[X] =\sum\limits_{n=0}^\infty \mathbb P(X>n) = \sum\limits_{n=1}^\infty \mathbb P(X\ge n)$. Curiously this change does not apply to the integral equivalent $\mathbb E[X] =\int\limits_{x=0}^\infty \mathbb P(X>x)\, dx =\int\limits_{x=0}^\infty \mathbb P(X\ge x)\, dx$ |
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1d
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Is the variance of the mean of a set of possibly dependent random variables less than the average of their respective variances? That looks as if it is addressing $\text{Var}(\bar{X}) = \frac{1}{n^2}\sum_{i=1}^n \text{Var}(X_i)$ rather than $\text{Var}(\bar{X}) \leq \frac{1}{n}\sum_{i=1}^n \text{Var}(X_i)$ (which is true, with equality only when $P(X_i-E[X_i]=X_j-E[X_j])=1$ for all $i,j$ i.e. when there is perfect positive correlation and identical variances). |
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1d
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Is the variance of the mean of a set of possibly dependent random variables less than the average of their respective variances? Cross post of math.stackexchange.com/questions/4942864/… which has four upvoted answers |
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1d
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revised |
In a sum of high-variance lognormals, what fraction comes from the first term? more persuasive |
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1d
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How do these books (about the German resistance against Hitler) differ? @SteveBird (2) may have a softer cover than (1) |
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1d
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Is the variance of the mean of a set of possibly dependent random variables less than the average of their respective variances? @GregMartin it is well known. I have added an additional introductory line |
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1d
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Is the variance of the mean of a set of possibly dependent random variables less than the average of their respective variances? covariances |
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1d
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answered | Is the variance of the mean of a set of possibly dependent random variables less than the average of their respective variances? |
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1d
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Finding out the Probability Mass Function from a Probability Generator Function? Your last expression looks like a natural consequence of your penultimate expression, at least when the support of $X$ is (a subset of) the non-negative integers. All the other terms of $G_X(s)$ either are differentiated away, or become $0$ when evaluating at $s=0$. |
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1d
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Should one account for the known variance of fixed X when estimating its relationship with random Y? So you are saying that the $(X,Y)$ are sampled. I would still suggest you want to base your analysis on the $X$ actually seen rather than what they might otherwise have been; you do not have $Y$ observations for those unseen $X$. |