Probability inequality of an average in terms of individual probabilities

Let $ X_1, cdots, X_n $ be identically distributed non-negative real-valued random variables that are not necessarily independent. Note that we have trivially for everything $ varepsilon> 0 $:

$$ mathbb {P} left { frac {1} {n} sum_ {i = 1} ^ {n} X_i geq varepsilon right } leq sum_ {i = 1} ^ { n} mathbb {P} {X_i geq varepsilon } = n mathbb {P} {X_1 geq varepsilon }. $$

For $ n geq 3 $, is there still an example where the above inequality holds for equality, or can we get a sharper bound for $ n $? I am particularly interested in the framework where $ n $ is wide.

For $ n = $ 2, the example $ X_1, X_2 sim U [0,1] $ with $ X_2 = 1 – X_1 $ and $ varepsilon = 0.5 $ presents an example where inequality is about equality, but it seems difficult to generalize to $ n> 2 $.

usability tests – In-depth individual interviews vs co-design workshop

In my current project, we have planned a usability test component using the method of in-depth individual interviews (IDI).

Due to unforeseen circumstances, the project must be delayed. I wonder if we can replace a co-design workshop rather than IDI because we do not want to extend the time further.

Here are my considerations:

Usage tests

  • We have around 4-5 modules which require different participants
  • In total, we will have to test with about 20-25 participants
  • It will take at least a month to complete it
  • However, IDIs will help us to identify navigation issues as well as mismatched interface issues

Co-conception workshop

  • We will bring together at least 5 users of each module (each 1 session module)
  • At most, it will take less than 2 weeks
  • We will print the designed wireframes and paste them in the room and bring users to discuss / work on them to validate the content
  • However, users will not be able to navigate the system, so I am concerned that we may experience usability issues. But they are able to validate the content (for example, form fields, information, statuses, etc.)

Let me know if it is a good idea / possible to make such arrangements. Thank you!

How do you calculate the contribution of each individual number to the product?

If you have four digits, say 1.1, 1.02, 0.98, 1.8 and the product of these are 1.979208, how do you standardize the contribution of each individual number to the product?

In case of addition, the sum is 4.9 so the contributions would


Calculated as (for example)

1.1/4.9 = 22.4%

I hope that the contribution of 0.98 should have a negative contribution on the product, just like a negative number on the sum.

wordpress – Some individual posts do not open correctly in a WP on AWS Lighsail

I have a WordPress no Lightsail of AWS with many messages, some messages do not open properly, it does not load auxiliary files like Images, CSS and JS, I discovered that it extracts these files from the machine's IP on the AWS and not from the domain, the ip does not have an SSL certificate and the files do not load because they are not secure. If I change the permanent link of the defective message, it opens normally, and if I try to access the message on the old permanent link, it does not give an error message not found, it opens again without external files.

I have a cache plugin installed WP Fastest Cache.

Anyone have any ideas or has there been a tip?

root access – Individual commands work fine, but shell script generates errors

I am trying to create a shell script to configure iptables and add some routes to my device running Android 10. I have root access.

Here is the script. I called him

iptables -t filter -F FORWARD
iptables -t nat -F POSTROUTING
iptables -t filter -I FORWARD -j ACCEPT
iptables -t nat -I POSTROUTING -j MASQUERADE
ip rule add from lookup 61
ip route add default dev tun0 scope link table 61
ip route add dev wlan1 scope link table 61
ip route add broadcast dev wlan1 scope link table 61

sysctl -w net.ipv6.conf.wlan1.disable_ipv6=1

I have a root console open via ADB, and I can execute each individual command from the script manually in the console, as follows:

enter description of image here

… etc.

But this happens when I run my script:

enter the preformatted text here

and I don't really understand why.

I have limited skills with Android and Linux, so I tried to follow and combine a few examples, but I was unlucky.

Can noindex tags on individual articles affect global SEO or ranking on Google?

Some SEO gurus promise that noindex attributes are good for devaluing unimportant pages. But when I asked in the Q&A moz, they are quite sure that most of the noindex article on the website can hurt the overall SEO ranking. What is your opinion on this.

game design – How to make the enemy appear in individual rows, by scrolling down the screen [using 2D DirectX,C]

I am a beginner in game development and I just started creating one 6 months ago. I'm creating a top-down 2D scrolling shooter using DirectX9 and C language. (I know it's an old one to use but I don't have a choice because it's a school project).

The result I want:

  • Enemies will be paraded in endless rows from the top to the end of the screen

The problem I have:

Currently, I have managed to spawn enemies in a single row. I don't know how to cut them and sort them into 8 enemies in each row. I really need to know how to sort this out and I hope someone will help me understand this !!

Thanks in advance.

Multiple selection to act or individual action buttons

I have a problem with two different types of action models. My approach was multiple line selection and a single action button at the top. Another approach is to have an action button on each line, which is not visually appealing. But there is always a pre-action (required) which must be done before clicking on the action.enter description of image here

Google sheets – on the edit script for time stamping on multiple columns with individual time stamp based on the time of the change made, all having the same time stamp

I have checkboxes for all the other columns starting with column C and I want the date stamp to appear in column d of row 2. The script works fine for this, but I also have need to be for column e (check box) Column f Fills a date timestamp; column g (checkbox) column h fills the date timestamp. Originally, I found this to stop the automatic updating that Google's sheets do every day. Any help would be greatly appreciated!

function onEdit3() {
var s = SpreadsheetApp.getActiveSheet();
var r = s.getActiveCell();
  if( r.getColumn() != 8 ) 
var row = r.getRow();
var time = new Date();
time = Utilities.formatDate(time, "GMT+05:30", "dd/MM/yyyy, hh:mm:ss");
SpreadsheetApp.getActiveSheet().getRange("h" + row.toString()).setValue(time);

How to test individual speakers on a MacBook Pro?

The 2019 MacBook Pro has 6 built-in speakers. Is there a way to test them 1 at a time? All I can find is how to test the left side or the right side independently, or how to run Apple Diagnostics which, I'm not convinced, will detect the degraded sound from one of the speakers .