probabilistic algorithms – Obtaining an expectation in uniform hashing

long shot question but I am super stuck.

Donald Knuth has proven (p. 8 here, equation 12) that the probability that the maximum value in uniform hashing is smaller than $$n/2$$ is equal to 0.288. I wonder if with this information I can recover what is the expectation of the maximum value? Simulation strongly suggest 0.63 n but I would like to understand what I am doing.

positive definite – Is the expectation of the inverse of a random matrix that has diagonal expectation also diagonal?

Suppose we have a symmetric positive definite random matrix $$mathbf{A}$$ that has a diagonal matrix expectation,
$$mathbf{E}(mathbf{A}) = text{diag}(mathbf{x}),$$
(like e.g. the Wishart distribution with diagonal scale matrix, $$mathcal{W}(text{diag}(mathbf{x}),n)$$).

Is it true that
$$mathbf{E}(mathbf{A}^{-k}),$$
where $$k$$ is an arbitrary positive integer is also a diagonal matrix?

user expectation – Why google material design has Import icon which is used as export in other design system?

When my teammate and myself were discussing the Import option for our application which adopts Material design Icons & Components.

We founded that there is a slight variant of the Import Icon is being used in Material Design System when compared to other design systems.

Material Import Option used in Google Contacts

IBM uses a similar icon for Export

Originally we started the discussion on the Import icon, but in the mid, we got confused with the Export option as well.

Which arrow do we need to use as a best practice for Import and Export?

user expectation – How do I best make a tally of items on a mobile phone

I’m designing a wireframe for a mobile app for logging materials.
One of the features needs to be able to log the makeup of a material.

fx in the case of a technical fabric the composition can look something like:

30% WO, 30% CO, 20% AC, 10% PU, 10% WS

tallying to 100%

My main thinking sofar has been making an add button at the button at the bottom of the screen and then have the user add each individual part until the full composition is reached.

However where I’m unsure is how to make sure the full tally is 100% – not more not less, what sort of feedback?
But also how I would lay this out on the phone screen if I use a list and an add button?

Does anyone know of any apps that deals with something similar or know best practise for something like this?

probability – Expectation of the minimum of two continuous random variables – using the joint pdf

Define $$Z = min(X, Y)$$ and the joint pdf of $$X$$ and $$Y$$ as $$f_{XY}(x,y)$$.

I saw an approach that said

$$E(Z) = int int min(x,y) f_{XY}(x,y) dydx$$

Is this readily obvious, or do you need to convert the following:
$$E(Z) = int min(x,y)f_Z(z) dz$$

to the above?

Expectation of an exponential of a random matrix

Suppose an $$n-$$ dimensional, real square matrix $$S$$ is a linear function of random variables with bounded variance (and possible higher order moments). Is it possible to find another $$n-$$ dimensional, real square matrix such that $$mathbb{E}(e^S) = e^D?$$

user expectation – How to design schedule service interface

I suppose is a big question.

I am trying to create understandable and clear UX design for schedule designer web app. May be, In a fact i’m looking for some well-known practice or production-ready interface, which fits our goals too.

So I think I have to describe our web app entities. To make it easier to understand my question I change names some of them to more usial.

So, imagine we have categories of followers and marketing emails to send. Also we have a schedule: each day we are going to send some pool of emails of some category.

Here is data model:

Also, I have to notice, that sometimes, we need to move some email of some category to another day of schedule, so
emails of day ≠ emails of followers category, even that looks like it.

What I decide to design right now:

The main key of this screen is `Generate schedule` button. We suggest that user needs to arrange categories to days and generate schedule from this screen. If user needs to move some email to another day so user could do it on another screen, when viewing generated content, and this idea looks bad.

I feel like I’m trying to reinvent the wheel. What is the most right pattern in this case? What already existed decision could fit my goals? Where to find right inspiration to get this done?
Any ideas or questions are welcome…

UPD.
The most problem is that user need to pre-generate schedule, and move some emails to another day if needed. But what if user wants to change schedule and generate one again? All corrections of previous version going to be lost. I look for decision where user could create “ready to use” schedule on fly.

probability – Conditional expectation based on two bivariate normal RVs?

Suppose $$X$$ and $$Y$$ are bivariate normal, both with mean 1 and sd 1. Considering the correlation $$rho$$, what should $$E[X|X,Y]$$ be? I think the answer should be just the RV $$X$$ as we conditioned $$X$$ itself. But is it possible that, since $$Y$$ and $$X$$ are correlated, it has some effect on the conditional expectation?

Thanks!

probability theory – Doob-like inequality with contional expectation

Suppose $$X_n$$ is a positive martingale w.r.t. a filtration $$mathcal{F}_n$$ such that $$X_n to 0$$ a.s. when $$n to infty$$. Given $$X^* = sup_{n in mathbb{N}} X_n$$, I want to prove that, for a given $$x in mathbb{R}^+$$
$$begin{equation*} mathbb{P}(X* geq x mid mathcal{F}_0) = 1 land frac{X_0}{x} end{equation*}$$

Where $$a land b = min{(a,b)}$$

This is very similar to the Doob martingale inequality, and, to get rid of the conditional expectation, I have tried to work with the sequence $$I_{B} X_n$$ with $$B in mathcal{F}_0$$ and prove it like Rosenthal;s A First Look at Rigorous Probability Theory proof of Doob’s inequality (Theorem 14.3.1):

Let $$A_k = { X_k geq x , X_i < x ;, ; forall ; 0< i . Note that $$A_i cap A_j = emptyset$$ . So

$$A = cup_{k=0}^infty A_k = { X* geq x}$$

Then

$$begin{split} x ; mathbb{P}(A|B) &= x sum_{k=0}^{infty} mathbb{E}({I_A I_B}) \ &= sum_{k=0}^{infty} mathbb{E}(x {I_A I_B}) \ end{split}$$

But I keep stuck there. ¿Is this the right path?