# Tag Info

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### Examples of distributions with easily solvable quantile functions but hard to solve CDFs

For a book-length answer to your question, see Statistical Modelling with Quantile Functions by Warren Gilchrist. In the following we assume random variables are continuous, let $F$ the the cdf of $X$,...

### Examples of distributions with easily solvable quantile functions but hard to solve CDFs

Assuming you mean evaluate rather than solve*, the Tukey lambda distributions have easily evaluated quantile functions but the cdf doesn't have closed form https://en.wikipedia.org/wiki/...

### Can a non-symmetrical distribution have the same areas under the PDF in the two sides around the mean?

A nice example is that any Binomial distribution with integer mean has median equal to its mean (this is hard to prove in general, but easy in any specific example).
• 41.9k
Accepted

### Can a non-symmetrical distribution have the same areas under the PDF in the two sides around the mean?

Your title question is identical to asking "Can an asymmetric distribution have mean equal to median" to which the answer is "yes" and many examples are to be found on site already....
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### Estimating mean and SD given the median and IQR values

The long answer is "no" this is not possible to determine without additional information about the distribution of the data. The pragmatic answer is "yes", if you make assumptions ...
• 5,654
Accepted

### Uniform prior and poisson likelihood, what posterior distribution will be produced?

Explicitly for the Poisson distribution: With likelihood $$f(x|\lambda) = \frac{\lambda^x e^{-\lambda}}{x!}$$ and prior $$f(\lambda) = \frac{1}{b-a} \textbf{1}_{a \leq x \leq b}$$ where $\textbf{1}_A$ ...
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Accepted

• 82.4k
1 vote

### Distribution to describe size of population with random exponential growth

It is not an orthodox approach to population modeling, because you are not estimating the usual logistical model parameters, and I guess the cells are immortal in your assays. But if you express the ...
• 63.7k
1 vote

### Calculating the joint pdf of linearly dependent random variables $X$ and $Y=X$

As already explained in comments, there is no joint density in the plane, because all the probability mass of $(X, X)$ is concentrated on the diagonal $y=x$. There is a density on that diagonal, but ...
• 81.4k
1 vote

### Unit-Root Asymptotics

It would appear (but check), that your case is, or is close to, Hamilton's book, Case 3, pp. 495-497, although he does not use the differenced dependent variable (his specification is his eq. 17.4.14)....
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1 vote

### How to fix intersection of cluster distributions in R

I don't see why you are using cluster analysis (CA). CA is unsupervised learning. Its goal is to find ways that observations "go together" or cluster, with no dependent variable. So, it is ...
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1 vote

### How can I find the limiting distribution of $Z_n=\sqrt{n}\frac{X_1X_2+X_3X_4+\cdots+X_{2n-1}X_{2n}}{X_1^2+\cdots+X_{2n}^2}$?

Let $X_1, X_2, \cdots$ be i.i.d. random variables with $E(X_i) = 0$, $\text{Var}(X_i) = 1$, and $E(X_i^4) < \infty$. We are tasked with finding the limiting distribution of  Z_n = \sqrt{n} \frac{...
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