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2 votes
Accepted

Mean of geometric distribution is odds?

If $p$ denotes the probability on success then $n$ trials will contain approximately $np$ successes and approximately $nq$ failures (where $q:=1-p$). So against $1$ success there will stand $\frac{nq}{...
  • 386
1 vote

what is the difference between mixture of two normal distributions and sum of two independent variables

Adding another answer to the mix, in case it helps. The random variable $Z = X + Y$ has a CDF of $P(Z\le z) = P(X+Y\le z)$. On the other hand, if you have a mixture distribution $Z\sim 0.5X + 0.5Y$, ...
  • 117
1 vote

Does this distribution belong to exponential family and does its support depend on $p?$

As @whuber says, what you describe is a mixture model. It looks like $Y \mid D=1 \sim \text{Exp}(\text{mean}=\theta_1)$ and $Y\mid D=0 \sim \text{Exp}(\text{mean}=\theta_2)$, where $D \sim \text{...
  • 9,569
0 votes

Is there a distribution for use with generalized linear models that captures both heavy tails and "pointiness" near the mean?

You can not use GLM To solve a generalized linear model (GLM) one uses a iterative algorithm that computes an ordinary least squares problem while changing the weights and values of the observations (...
5 votes

Is there a distribution for use with generalized linear models that captures both heavy tails and "pointiness" near the mean?

You can't interpret the shape of the residuals without checking the conditional mean and variance assumptions (e.g. by residuals vs fitted); if the model for the conditional mean was wrong or the ...
  • 264k
1 vote

For ecological data, when is a gaussian distribution appropriate?

In such cases, you may want to use a non-parametric test, like the Kruskal-Wallis test.
  • 662
4 votes
Accepted

Which distribution is it?

It's unusual. Let $\beta = 1/a$ and $Y = 1/X$ (supported on the interval $(\beta,\infty)$) so that, exploiting the continuity of this distribution, $$\Pr(Y\le y) = \Pr(X \ge 1/y) = \Pr(X \gt 1/y) = 1 ...
  • 297k
2 votes

Is there a distribution for use with generalized linear models that captures both heavy tails and "pointiness" near the mean?

The first thing that comes to mind is the double exponential distribution. It looks like there's a nimble package that might help (rdocumentation.org/packages/...
0 votes
Accepted

Calibrating the probabilities of Ridge Classifier on imbalanced dataset

The problem is that the histogram of probabilities show that there is no separation between the 2 classes, with an almost normally distributed density With an AUROC of 0.76, I think this is expected. ...
  • 3,231
1 vote

Does a misspecified model always have lower likelihood value than the correct model?

I am going to slightly rephrase your question: we assume you have $N$ samples $\{x_i\}_{1 \leq i \leq N}$ which were generated from a ground-truth model $d_1$ with parameters $\theta_1 \in \Theta_1$ (...
1 vote

Distribution of difference of two random variables with chi-squared distribution

Using the following links and parameter correspondence between the characteristic function(chf) of difference of $\Gamma(\alpha,\nu_{\Gamma})$ and $VG(\sigma,\nu)$ symm. variance gamma rvs we can ...
1 vote

Distribution of $x_4(x_1-x_3)+x_5(x_2-x_1)$ with iid $x_i \sim N(0,1)$

An almost exact expression could be derived using link between $\chi^{2}$, $\Gamma$ and symmetric $VG$ variance gamma distributions. Given very useful results above by @whuber, we can proceed first by ...
0 votes

Distribution check using R

Much useful information in the comments here, but I must add a point overseen by the commenters: You are trying to compare the fits by looking at their log-likelihood values. In R this is simply ...
0 votes

Are there better measures of entropy

Variance can be generalized to be about any point rather than just the mean (i.e. how spread out is the distribution from this point) $$\mu \mapsto\sum p_i (x_i - \mu)^2.$$ We can also "invert&...
1 vote

Are there better measures of entropy

Shannon's entropy distinguishes between those two scenarios: Two bins: $$- \sum_{i=1}^2 \frac{1}{2} \log \frac{1}{2} = \log 2$$ Ten bins: $$- \sum_{i=1}^{10} \frac{1}{10} \log \frac{1}{10} = \log 10$$ ...
  • 4,952
2 votes

Alternative formula for the Bernoulli pmf?

This is fine, assuming that the domain of $f$ is $\{0,1\}$. This is also true of the formulations in the other answers. A different formulation involving the Iverson bracket is \begin{align*} f(x) = (...
  • 129
3 votes
Accepted

What is the distribution of bit counts of a binomial random variable?

We have a well-known distribution $X \sim B(n,p)$, i.e. we know $p(x)$, and a well-defined function bit_count: $x\to y$ for the domain $x\in [0:n]$. Then, the ...
  • 8,369
9 votes

Alternative formula for the Bernoulli pmf?

There's nothing wrong with it as it evaluates the values, it should evaluate. The usual formulation however uses powers so it becomes a case of binomial distribution with $n=1$ sample size. Recall ...
  • 121k
0 votes
Accepted

Calculate the variance of a distribution analytically

You appear to be asking for the marginal distribution of $X$ where $(X,Y)$ has a uniform distribution on a sheared unit square. (The unit of measurement is the base of the square.) Because the ...
  • 297k
7 votes
Accepted

Alternative formula for the Bernoulli pmf?

Your alternative form is often written braced form as $$ f(x)=\begin{cases} p & \text{if $x=1$} \\ 1-p & \text{if $x=0$} \end{cases} $$ and there is nothing wrong with ...
0 votes
Accepted

To check if the churn probability score from old and new model is similar

Because the two models are ostensibly the same, just with different sourcing of the variables, I would just plot the predicted probabilities against one another and look for large discrepancies. You ...
0 votes

Binomial to Poisson Approximation

Going by Prof G E P Box's quote "in statistics no model is perfect but some are useful", modeling data using Probability Distributions also fits this quote very well, one can use any ...
0 votes

Calculate the variance of a distribution analytically

I think your h(x) is a so called mixture distribution. In your example, there are actually three distributions of h(x), either h1(x)=ax, h2(x)=c, or h3(x)=-x+w, with h1(x) and h3(x) both being uniform,...
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0 votes
Accepted

Binomial to Poisson Approximation

That should be $np\geq 5$ and $n(1-p) \geq 5$ for any approximation to be taken on Binomial Distribution. The approximation can be attributed to Central Limit Theorem. This type of approximation is ...
0 votes

Calculate the variance of a distribution analytically

Your distribution seems to be a Trapezoidal distribution; analytical expressions for its different modes can be found on the following page: https://en.wikipedia.org/wiki/Trapezoidal_distribution Hope ...
2 votes

Distribution function of an exponential random variable

Although this is a self-study question, since it is now is over five years old, I'm going to go ahead and give a full answer for expository purposes and to assist later users. Let's generalise your ...
  • 102k
1 vote

Is there a test that can tell me, even for low counts, the probability that the obtained distribution of values eg from a die is truly random?

Test for goodness of fit with categorical data, whether a particular distribution is a good fit for a particular sample, are the Pearson's chi-squared test and the G-test. Note that those tests are ...
1 vote

How to estimate the probability mass function of a discrete variable from moments

I have worked with this problem a lot, and I cannot find a satisfactory answer, but here are a few ways that I've attacked it. (1) If $M=k$, and if you know that a valid probability distribution can ...
  • 11
1 vote

KL divergence between gaussian and uniform distribution

How else can I calculate the distance of my gaussian to a 'maximum entropy' distribution if I can't use the uniform distribution? You basically answered your question, you can use the entropy of the ...
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3 votes
Accepted

PDF of $Z=X^2 + Y^2$ where $X,Y\sim N(0,\sigma)$

Following the comment by whuber, for problems like this that involve convolutions of IID random variables, it is generally simpler to work with the characteristic function than with the density ...
  • 102k
0 votes
Accepted

Maximum likelihood of Normal density under selection

To simplify the analysis, I will first deal with the special case where $\sigma=1$. Your likelihood function here can be written by removing the proportionality constants, which gives: $$\begin{align}...
  • 102k
1 vote

Extreme value distribution for multivariate normal

Bowler, This is an interesting question that I see is unanswered. Memming suggested the Rayleigh distribution, which would work for two Axis (Plane) Radial errors Rxy Rxz Ryz from x,y,z Normally ...
1 vote

Why does central limit theorem give such big x in $\phi(x)$

The set up of the question is absurd, as originally pointed out by Michael M, in that the expected number with $200$ days is $10000$ with a standard deviation of $100$, so you are extremely unlikely ...
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1 vote
Accepted

Why does central limit theorem give such big x in $\phi(x)$

You're looking at a case where (assuming independence, though it would seem to be a somewhat questionable assumption), the distribution of the number of visitors in $200$ days has $\mu=10000$ and $\...
  • 264k
2 votes
Accepted

Estimating the average building size

Suppose, the number of people living in a random building of the city (chosen uniformly) is $X$ and has probability mass function $P(n)$. Now the number of people living in the building where a random ...
0 votes
Accepted

How to formulate the distribution and conduct a hypothesis test on the following situation?

Your intuition about the limit distribution of the longest consecutive streak of heads is correct. There is a theorem, proved by Antónia Földes in "The limit distribution of the length of the ...
2 votes
Accepted

Using KDE to approximate a Price vs Quantity curve

The KDE is a nonparametric method, thus you don't provide any domain knowledge to the model, other than some degree of smoothness via the bandwidth. If you do have any domain knowledge, like e.g. that ...
  • 8,369
1 vote

Time distribution: is it possible to determine whether calls are human-made or machine-made by patterns of the distribution

Over the short term the answer is maybe. If we assume yes, then as soon as it becomes possible to model human-made calling patterns versus machine-made calling patterns, that very model can be used to ...
  • 27k
1 vote
Accepted

What information can I extract from an overlap of two personal probability distributions?

The ratio of two gamma-distributed RVs is the generalized beta prime distribution. The CDF can be defined in terms of the ordinary hypergeometric function, ${}_2F_1$: $F(x;\alpha,\beta,q)=\frac{(xq)^{\...
  • 656
0 votes
Accepted

Erroneous Argument for uncorrelated implies Independence

Note: you can safely work with the pdf here since we know it exists For $(\Rightarrow)$, when replacing $\text{cor}(X,Y)=0$ in the expression of $f_{X,Y}(x,y)$, do you see that the joint density ...
  • 1,489
2 votes

What should be the formal definition of continuous uniform distribution pdf value at upper bound?

Partially answered in comments: No pdf is defined, as a function, at any point. As a convention, a pdf is often represented as a function that is continuous wherever possible. Because no pdf for ...
0 votes
Accepted

Comparison between multiple curves/probability distributions

Once you have your pairwise distances (another possibility is the Earth Mover's Distance, also known as the Wasserstein metric), you can cluster your sets. The simplest clustering algorithm would be ...
3 votes
Accepted

Continuous and differentiable bell-shaped distribution on $[a, b]$

Let's construct all possible solutions. By "distribution" you appear to refer to a density function (PDF) $f.$ The properties you require are Supported on $[a,b].$ That is, $f(x)=0$ for ...
  • 297k
3 votes

Continuous and differentiable bell-shaped distribution on $[a, b]$

The Truncated normal distribution obeys all prerequisites: It's bell shaped It's continuous Its support is $x \in [a,b]$ It's differentiable, i.e. $\nabla_x p(x)$ exists for all $x \in [a,b]$
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3 votes

Continuous and differentiable bell-shaped distribution on $[a, b]$

One option is to transform a beta distribution. $Beta(3,3)$ has your desired properties on $[0,1]$. Now subtract $1/2$ to center the distribution. Next, multiply to stretch or compress the ...
  • 35.7k
1 vote
Accepted

Create statistic to judge moisture distribution 'quality'

You are juggling a lot of balls there. The five features you require of a "good" distribution already are nontrivial to operationalize all by themselves. "Low moisture" can refer ...

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