probability or statistics – Adding a module to teach Mathematica the Laplace transform of particular Mittag-Leffler functions

Math. 11.3 is not aware of a useful Laplace transform

LaplaceTransform(t^(-a) MittagLefflerE(a, a, t^a), t, s)

which for 0<a <=1, Re(s)>1 should be

(s^a - 1)^(-1)

as easily checked

s = 2; a = 1/2; Print(
 NIntegrate(
  Exp(-s t) t^(-a) MittagLefflerE(a, a, t^a), {t, 0, Infinity}), "=",
 (s^a - 1)^(-1) // N)

(a similar remark was made in Mittag Leffler function Laplace transforms with Mathematica )

I guess it should be possible to add this simple rule via a module. This would allow for example inverting Laplace transforms which are rational functions of s^a, which may be one of the most basic applications of these particular Mittag-Leffler functions.

forms – Has anyone statistics for the number of users completing a registration for a service?

Majority of the advertisers face issues when it comes to starting new ad campaigns, which are not attracting enough clicks and traffic to their site. The problem with low traffic is that if you’re not working towards qualified traffic, as per the strategies of various marketing agencies, including Point Blanc. You will not lead or sell the campaigns, despite the implementation of keywords thoroughly.
So, if you have ten clicks in one day on your campaign, you will be able to do four sales in one day. However, if we increase the clicks to double, keeping a 40% conversion rate, the leads will be eight a day.
A campaign without any traffic is like dead fish. So, here are some tips to make it better!

Budget and Bids
Google ads experts in UAE claim that this is the easiest measure that enables you to increase clicks. When the campaign bids are increased, the ranks of the ads are higher, leading to a higher average result on Google’s search result page.
Your campaign ads will be more visible when you are in a higher position, increasing the potential clicks as well. However, keep in mind that campaigns are contingent on the budget. So, increasing just the bids will reduce the number of clicks as the CPC average increases. Moreover, the clicks will also be fewer for the same budget on a daily basis.

Target Areas
Mostly, PPC wants to start small as a test. However, this timid approach would not get them the potential lead generation. Increasing the target area will enable you to drive more traffic as compared to being confined to one specific area.

Keyword Match
The best way to deal with this is through a broader keyword match. Broad match modifiers can also be tried on some of the main ad groups of the campaigns. It usually drives more traffic in contrast to Phase or Exact Match, which would help in boosting the total click volume.

statistics – Is the formula for standard error for the slope of a linear regression with intercept the same as without?

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probability or statistics – Modelling GBM with ItoProcess function

I am trying to model GBM using ItoProcess function. I am not sure why I am getting negative values in the output for GBM.
I am not using GeometricBrownianMotionProcess function because I want to model other SDE also.

r = 33;
s = 8;
sde = ItoProcess[[DifferentialD]z
r z
z

sol = RandomFunction[sde, {0, 10, 0.01}];
ListLinePlot[sol,   Filling -> Axis]

enter image description here

statistics – How do I graph the chance of success with a range of dice pools against a range of target numbers?

I’m having a hard time coding an anydice script to show what I want.

Let me contextualize the mechanic I’m trying to simulate:

You roll a POOL of d10 against a TARGET number. If at least one die from the POOL is equal to or higher than the TARGET, the roll is a success. The count of such dice is the degree of success, but that isn’t my focus at the moment.

I’d like to have a graph for the chances of success of various POOLs of different sizes up to 10 (1d10, 2d10, 3d10…10d10) against different TARGETs from 2 to 10 (2, 3, 4… 10).

The caveat is: I’d like the graph to be layed out in such a way that:

  • the x axis represents the TARGETs;
  • the y axis represents the chances of at least 1 success;
  • each line represents a POOL,

so I can see the chances that each POOL has to succeed against a whole range of TARGETs.

Can any anydice wizard help me with this, please?

statistics – ANYDICE – Help with a dice pool showing success against a range of target numbers

I’m having a hard time coding an anydice script to show what I want.

Let me contextualize the mechanic I’m trying to simulate:

You roll a POOL of d10 against a TARGET number. If at least one die from the POOL is equal to or higher than the TARGET, the roll is a success. The count of such dice is the degree of success, but that isn’t my focus at the moment.

I’d like to have a graph for the chances of success of various POOLs of different sizes up to 10 (1d10, 2d10, 3d10…10d10) against different TARGETs from 2 to 10 (2, 3, 4… 10).

The caveat is: I’d like the graph to be layed out in such a way that:

  • the x axis represents the TARGETs;
  • the y axis represents the chances of at least 1 success;
  • each line represents a POOL,

so I can see the chances that each POOL has to succeed against a whole range of TARGETs.

Can any anydice wizard help me with this, please?

statistics – What is the motivation of naming “Marginal Likelihood”?

I gathered follwing remarks from internet –

  1. In statistics, the margin (as in “marginal distribution”) is the average or, in mathematical terms, the integral.

  2. Marginal means that we marginalised, integrated, the variable $theta$

  3. A marginal likelihood is the average fit of a model to a data set. More specifically, it is an average over the entire parameter space of
    the likelihood weighted by the prior.

  4. This marginal likelihood, is just the normalizing constant of Bayes’ theorem.

The main query is what is the meaning of marginal likelihood which I asked in this post (click here),

though the answer is not satisfactory I get the idea, that marginal likelihood, is just the normalizing constant,

it sums all the probabilities for all respective case or data, so I understand that this is some kind of integration.

But then why name it “marginal”? marginal means “relating to or at the edge or margin”, how expresses summation or integration?

Also, why I find in the internet that marginal means average? Plz explain, thanks.

statistics – The posterior distribution of the Poisson-Gamma Model

I am trying to figure out why the following holds true and would like to ask for your help.

Given the following Bayesian model:
$$
y_i sim text{Poisson}(mu_i theta_i)\mu_i sim text{Gamma}(alpha, beta) \alpha sim text{Exponential}(a)\beta sim text{Gamma}(b,c)
$$

for available observations $y_i$ with $i=1,…,n$ and parameters $theta_i, a, b, c$ fixed the posterior distribution of the parameters $mu_i, alpha, beta$ is:

$$
pi({mu_i}, alpha, beta | {y_i},{theta_i},a,b,c) propto prod_{i=1}^n (f(y_i | mu_i, theta_ipi(mu_i|alpha, beta))pi(alpha|a)pi(beta|b,c)
$$

How is this equation derived?

statistics – 90% confidence interval for mining a block?

Bitcoin blocks are mined according to a Poisson process with a mean of 10 minutes (600 seconds) if you assume constant network hash rate. The difficulty of mining blocks gets adjusted every 2016 blocks to address this variability.

Interarrival times of a Poisson process are exponentially distributed.

The 95th quantile of an exponential distribution with a parameter of 1/600 is 1797.4

The 5th quantile of an exponential distribution with a parameter of 1/600 is 30.8.

90 percent of the time an observation from an exponential distribution will fall in the interval of (30.8,1797.4) seconds

The median time is 415.9 seconds.