12 votes
Accepted

What uncited method is being used to test for lead water pipes in the US?

This is a bit of a rabbit hole. It seems that the $357$ is the answer to a completely different question. Its calculation comes from the sample size for an infinite population of $$\dfrac{1.96^2 \...
Henry's user avatar
  • 38.7k
6 votes

How good is the Beta distribution as a conjugate for Binomial distribution?

I don't have my textbook in front of me, but I recall the Bayes estimator with conjugate prior is Bayes optimal meaning the posterior mode minimizes the MSE. In other words, it is a biased estimator ...
AdamO's user avatar
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5 votes
Accepted

How good is the Beta distribution as a conjugate for Binomial distribution?

The fact that there is an (arbitrary?) $c$ scaling up and down the posterior distribution makes me think that the Beta distribution ... may not be a good representation of the distribution of $\theta$,...
Sextus Empiricus's user avatar
5 votes
Accepted

Taking the limit of a Beta Distribution to yield the Gamma Distribution

Suppose $X\sim\operatorname{Beta}(\alpha,n).$ Consider the limit of the distribution of $nX$ as $n\to\infty.$ \begin{align} f_{nX}(x) = {} & \frac d{dx} \Pr(nX\le x) = \frac d{dx} \Pr\left( X\le\...
Michael Hardy's user avatar
4 votes

What is the standard error of a binomial process with a false-positive rate

Let $T$=TP and $F$=FP, which are known and constant. Also assume $T\neq F$. Taking a step back, let $Y_i$ be an independent binary variable corresponding to a single patient's test result. We have \...
cambridgecircus's user avatar
3 votes

Mann-Whitney U test, compared to Binomial test

Your $n_An_B/2$ comparisons aren't all independent, so their sum doesn't have a Binomial distribution; it has a distribution with larger variance than a Binomial, so your test assuming Binomial ...
Thomas Lumley's user avatar
3 votes

If $X \sim\textrm{ Bin}(100, 0.5), $ then what is the approximate distribution of $(X/5 - 10)^2? $

$X \sim \text{Binomial}(100, 0.5)$ can be approximated by a Normal distribution with a mean of $100 \times 0.5 = 50$ and variance of $100 \times (0.5) \times (1 - 0.5) = 25$. Therefore $\frac{X}{5}$ ...
Eoin's user avatar
  • 8,867
3 votes

What uncited method is being used to test for lead water pipes in the US?

A couple of references I found: ASTM standard Research Article with good references The state of Colorado paper seems to parallel this one from Michigan It references this group's work with this ...
R Carnell's user avatar
  • 5,123
2 votes

Taking the limit of a Beta Distribution to yield the Gamma Distribution

Consider the integral over the Beta Distribution: $\int^1_0Beta(x;\alpha,\beta)dx = \int^1_0\frac{(\alpha + \beta -1)!}{(\alpha -1)!(\beta-1)!}x^{\alpha-1}(1-x)^{\beta-1}dx=1$ Transforming variables ...
SSD's user avatar
  • 185
1 vote
Accepted

If X ~ Bin(2, 1/2) and Y ~ Bin(3,1/3), then find P(2X+3Y=13)

Given: $$X \sim \text{Bin}\left(2, \frac{1}{2}\right)$$ This means $X$ can take on values $0, 1, 2$ with probabilities following the binomial distribution formula $$P(X=k) = \binom{n}{k} p^k (1-p)^{n-...
ADAM's user avatar
  • 679
1 vote

Mann-Whitney U test, compared to Binomial test

The pairwise comparisons are not independent One way to consider the computation of the $U$ statistic is by making from the $n_1$ samples $X$ and $n_2$ samples $Y$ a number of $n_1\cdot n_2$ pairwise ...
Sextus Empiricus's user avatar

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