**5.538 distributions questions.**

I've got some data (158 cases) which was derived from a Likert scale answer to 21 questionnaire items. I really want/need to perform a regression analysis to see which items on the questionnaire ...

Given
$$X \sim \text{Beta}(\alpha,\beta)$$ (where $\alpha=\beta$, if that helps) and
$$\theta \sim \text{Uniform}(0, \pi/2).$$
I'm trying to find a formula for $P(Y)$ (or even the cdf) of
$$Y = X +...

Say we have $X \sim \text{Beta}(\alpha, \beta)$. What's the sampling distribution of its sample mean?
In other words, what distribution does the sample mean $\bar{X}$ of a Beta follow?

I have some data in the form of bins and counts. Here is one complete non-truncated example:
...

I'd like to create a reference sheet of common distributions for my statistical theory class, but I'm having some issues understanding R's implementation of the negative binomial distribution in the <...

I have a model that assumes a probability of survival over discrete time (example: decades)
...

What is the best way to check if implementation of density, distribution function, quantile function and random generation for some distribution are correct? For example, base R lacks Laplace ...

So I will have data sorted into one of five bins (flow cytometry sorting of cells). These bins are actually adjacent categories that quantify a single fluorscence level (0-20,20-40,40-60,60-80,80-100)....

I have learned about the intuition behind the KL Divergence as how much a model distribution function differs from the theoretical/true distribution of the data. The source I am reading goes on to say ...

Let we define $H(\theta)= 1- \sum^{y}_{k=0} {n\choose k}\theta^{k}(1-\theta)^{n-k} + \frac{1}{2}{n\choose y}\theta^{y}(1-\theta)^{n-y}$. Thus, $H(\theta)$ is CDF.
My corresponding ...

Recently I came across the following problem: Let $f:\mathbb{R}^2\to\mathbb{R}$ be a function as smooth as we want (even analytic if this is convinient) such that the equation $f(x,y)=0$ for every $x$ ...

I am wondering what could we obtain more from the predictive distribution.
Give a set of data, say $\mathcal{D}=\{(x_i,y_i)\}$, we want to predict the value s$Y_{new}$ at new locations, say $X_{new}$.
...

I am working on a choice model with Stata. Now I want to build an interaction term between a variable which is assumed to be log-normal distributed(price) and a variable which is assumed to be normal ...

I am working on a dataset and I got the following distributions, but I don't know how to interpret the distributions that I have got. Can you please explain me what are the conclusions from the ...

I have two lognormal distributions which represent the annual distribution of sales of fiction and non-fiction books, respectively. The sample size of fiction books is much larger than that of non-...

I have a signal (Y) with 200000 samples. I plotted probability density function (PDF) of the ...

Erlang Kernel is used for density estimation. By using this estimates are pretty close to the real density on graph on the other side MSE is very large. But Author of Erlang Kernel stated that it will ...

Brief question here. Kind of dumb. Say we have to independent normal distributions ($X$ and $Y$). What is the distribution of Z where Z is $(X + Y)^2$.
Thanks!

We know that the cumulative distribution function (CDF) follows the $U[0,1]$ distribution. What is the distribution of (1-CDF)? Is it also follows the $U[0,1]$ ? (I believe it's true for the normal ...

If a probability distribution has, say, 112 bins with around 29000 samples, with the maximum probability of a bin being less than 0.05, is the Jarque-Bera test an effective measure of conformance to a ...

I have a two dimensional constant vector $\mathbf{A} = \left < 2,1 \right>$. Also, I have a vector $\mathbf{e} = \left < \epsilon_x, \epsilon_y\right >$. Both $\epsilon_x$ and $\epsilon_y$ ...

The question that I wish to ask is what is an appropriate likelihood model to use for races.
For example, suppose that we have 4 competitors in a 100m race. We observe the competitors weight and age ...

Consider two sets of data points A and B. Both these data points are from mixture of unknown number of Gaussians. The mean of the Gaussians are little different for each set (there may have few ...

Given a distribution $d$ on non-negative numbers and a threshold $t > 0$, I define the "truncated" distribution $d_t$ where
$\left\{
\begin{array}{ll}
d_t(x) = 0 & \mbox{when} \ x ...

I need to simulate 365 million random numbers from a t distribution with given parameters in python. I need to simulate 1 million years each consisting 365 days and for each day I need a random ...

I am trying to measure the dissimilarity between two empirical discrete distributions. I am aware of various distance metrics that could be used for this purpose such as Wasserstein, Bhattacharyya etc....

In practice, we may always be asked to check the skewness and kurtosis of a data set. I have two questions.
Given a probability distribution, how can we determine/evaluate the skewness and kurtosis ...

I have stones of different weights. For each stone, I flip the same fair coin. If it's heads, I add the stone's weight to a running total. Given the weights, can I find the distribution for the total ...

There is a reference dataset which is a discrete count over a set of names which produces a distribution $P$, and a set of names that are entered into a website as usernames which has a distribution $...

Problem: You have a sequence of N steps that must occur in order. Each step is "unlikely" in any given time period (say, <10%). However, you know that all N steps happened to successfully ...

I have an event having poisson distribution with time intervals of one minute. Every event has accomplishment time with gamma distribution.
I $N$ number of events start in $t$ minutes, the what will ...

Sorry if this is a basic question but I can't find an answer that is clear enough, so I prefer to ask.
I want to model a number of events (number of gaze) that depends on the behaviour that one ...

I have $10000$ samples of 6-lettered strings of the following type
Left Right &...

We have a function P (t, x) that indicates a failure probability (for some experiment) for a given time t and a parameter...

Given the KL divergence value between 2 distributions, how should someone use this to determine whether the value is significant for the distributions $P$ and $Q$ to be different? One method I can ...

I'm trying to make sense of the following sentence from introduction "Multiple discoveries: Distribution of roots of determinantal equations"
http://statweb.stanford.edu/~ckirby/ted/papers/...

I have a problem where I am aware that data is well-modeled by a Zeta distribution such as $P(X=x) = x^{-a}/\zeta(a)$, and would like to learn the Zeta distribution parameter $a$ from the data. More ...

suppose $X_1,\ldots,X_n$ be random sample of continuous distribution with distribution function $F$ if $$Y=\prod_{i=1}^k F(X_i) \prod_{i=k+1}^n [1-F(X_i)]$$ how can I calculate distribution of $U=-\ln ...

First, I don't know which data to consider when I talk about the distribution of my data. All my data? Or each distribution of the data of all the levels of my conditions? I have tested people three ...

I have a hazard rate given by a 2-parameters Weibull distribution, in the form:
$$h(t) = \cfrac{B}{A} \, \left({\cfrac{t}{A}}\right)^{B - 1}$$
where $A=$ scale parameter and $B=$ shape parameter.
I ...

Suppose we are given a sample of data, x = (x1, ..., xn).
How to write the likelihood of this data, given that this is comes from a specific probability distribution with known parameters and the ...

I have plotted some experimental data of mine, and these data points fall into the following distributions:
So, these are fairly non-trivial looking distributions. I would like to figure out methods ...

I need to generate a hyperexponential distribution for my project. I have already implemented a poisson generating algorithm given by Donald Knuth, but I couldn't find an algorithm for generating a ...

I stumbled upon an odd result which I have difficulties to explain. In the following code, $x_1$ and $x_2$ are very similar variables. Yet the distribution of p-values for the coefficient in $x_1$ is ...

Background:
Suppose I have 2 random observations from a normal density function whose $\sigma = 2$. The first observation is $x_1 = 6$. The second observation is $x_2 = 8$. The question below arises ...

I am trying to fit some data using python scipy.fit with the gamma distribution.
This returns three parameters (alpha, loc, beta). I would like to use these parameters to generate some data in excel, ...

I have to divide a quantity amongst a known set. However, there are constraints and adjustments.
Example:
Quantity to share: 84
Divide amongst:
...

I want to quantify the complexity of the street network of different cities.
For each city I have the angle distribution of its streets.
My hypothesis that the more complex the street network, the ...

Let $X = (X_1, X_2,...,X_k) \sim \text{Multinomial}(n, \theta)$ on $k$ elements (i.e. each $X_i$ is the count of the element $i$ on $n$ trials). Let $\theta \sim \text{Dirichlet}(\alpha)$, where $\...

Let $X_1,X_2$ be i.i.d. $N(0,1)$ and $U_1,U_2$ be i.i.d. $U(0,1)$ and independent of $X_1,X_2$. Define $$Z_1=\frac{(X_{1}U_{1}+X_{2}U_{2})}{\sqrt{U_{1}^2+U_{2}^2}}.$$ Find the distribution of $Z$.
...

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