Menu
News
All News
Dungeons & Dragons
Level Up: Advanced 5th Edition
Pathfinder
Starfinder
Warhammer
2d20 System
Year Zero Engine
Industry News
Reviews
Dragon Reflections
Columns
Weekly Digests
Weekly News Digest
Freebies, Sales & Bundles
RPG Print News
RPG Crowdfunding News
Game Content
ENterplanetary DimENsions
Mythological Figures
Opinion
Worlds of Design
Peregrine's Next
RPG Evolution
Other Columns
From the Freelancing Frontline
Monster ENcyclopedia
WotC/TSR Alumni Look Back
4 Hours w/RSD (Ryan Dancey)
The Road to 3E (Jonathan Tweet)
Greenwood's Realms (Ed Greenwood)
Drawmij's TSR (Jim Ward)
Community
Forums & Topics
Forum List
Latest Posts
Forum list
*Dungeons & Dragons
Level Up: Advanced 5th Edition
D&D Older Editions
*TTRPGs General
*Pathfinder & Starfinder
EN Publishing
*Geek Talk & Media
Search forums
Chat/Discord
Resources
Wiki
Pages
Latest activity
Media
New media
New comments
Search media
Downloads
Latest reviews
Search resources
EN Publishing
Store
EN5ider
Adventures in ZEITGEIST
Awfully Cheerful Engine
What's OLD is NEW
Judge Dredd & The Worlds Of 2000AD
War of the Burning Sky
Level Up: Advanced 5E
Events & Releases
Upcoming Events
Private Events
Featured Events
Socials!
Twitch
YouTube
Facebook (EN Publishing)
Facebook (EN World)
Twitter
Instagram
TikTok
Podcast
Features
Top 5 RPGs Compiled Charts 2004-Present
Adventure Game Industry Market Research Summary (RPGs) V1.0
Ryan Dancey: Acquiring TSR
Q&A With Gary Gygax
D&D Rules FAQs
TSR, WotC, & Paizo: A Comparative History
D&D Pronunciation Guide
Million Dollar TTRPG Kickstarters
Tabletop RPG Podcast Hall of Fame
Eric Noah's Unofficial D&D 3rd Edition News
D&D in the Mainstream
D&D & RPG History
About Morrus
Log in
Register
What's new
Search
Search
Search titles only
By:
Forums & Topics
Forum List
Latest Posts
Forum list
*Dungeons & Dragons
Level Up: Advanced 5th Edition
D&D Older Editions
*TTRPGs General
*Pathfinder & Starfinder
EN Publishing
*Geek Talk & Media
Search forums
Chat/Discord
Menu
Log in
Register
Install the app
Install
The
VOIDRUNNER'S CODEX
is coming! Explore new worlds, fight oppressive empires, fend off fearsome aliens, and wield deadly psionics with this comprehensive boxed set expansion for 5E and A5E!
Community
General Tabletop Discussion
*Dungeons & Dragons
First playtest thread! One D&D Character Origins.
JavaScript is disabled. For a better experience, please enable JavaScript in your browser before proceeding.
You are using an out of date browser. It may not display this or other websites correctly.
You should upgrade or use an
alternative browser
.
Reply to thread
Message
<blockquote data-quote="Cadence" data-source="post: 8739309" data-attributes="member: 6701124"><p>and [USER=7031982]@Bill Zebub[/USER]</p><p></p><p>Ok, it was either this or do work I was supposed to do...</p><p></p><p>I think the easiest way to get the variances is to view it as a mixture distribution, where you have an R/N chance of having a discrete uniform from 1 to N (you rolled 1 to R the first time), and a 1-(R/N) chance of having a discrete uniform from (R+1) to N (you rolled a keeper the first time). That the mixture pretends you reroll the dice for keepers (to decide which keeper) doesn't matter since they have an equal probability either way.</p><p></p><p>So, putting the means and variances of the discrete uniform into the formulas for a mixture distribution I got:</p><p></p><p>m1=(1+N)/2</p><p>v1=(N^2-1)/12</p><p>w1=R/N</p><p></p><p>m2=(R+1+N)/2</p><p>v2=((N-R)^2-1)/12</p><p>w2=1-(R/N)</p><p></p><p>mmix=w1m1+w2m2</p><p>vmix=w1(v1+m1^2)+w2(v2+m2^2)-mmix^2</p><p></p><p>Insert algebra.</p><p></p><p>The mean comes out to be (RN+N+N^2-R^2)/2N, matching what you found.</p><p></p><p>The variance comes out to be gross, but just what you get by plugging things in. So we can calculate it pretty quickly for any given N and R. (I plopped those formulas right into R, and added taking a square root).</p><p></p><p>And the standard deviations are not what I was expecting!</p><p>[ATTACH=full]258483[/ATTACH]</p><p></p><p>It looks like when you get to rerolling the lowest half of the possible values that the variances start going up again, and can be larger than not dropping anything!</p><p></p><p>This seemed odd, so I calculated them out by getting the distributions in the d4 case, and they do look to be right.</p><p></p><p>[ATTACH=full]258482[/ATTACH]</p><p></p><p>Anyway, here are what I get for the ratios of the standard deviations for dropping 1s (and keeping the reroll) to the standard deviations of just keeping, for d3 to d100.</p><p></p><p>[ATTACH=full]258484[/ATTACH]</p></blockquote><p></p>
[QUOTE="Cadence, post: 8739309, member: 6701124"] and [USER=7031982]@Bill Zebub[/USER] Ok, it was either this or do work I was supposed to do... I think the easiest way to get the variances is to view it as a mixture distribution, where you have an R/N chance of having a discrete uniform from 1 to N (you rolled 1 to R the first time), and a 1-(R/N) chance of having a discrete uniform from (R+1) to N (you rolled a keeper the first time). That the mixture pretends you reroll the dice for keepers (to decide which keeper) doesn't matter since they have an equal probability either way. So, putting the means and variances of the discrete uniform into the formulas for a mixture distribution I got: m1=(1+N)/2 v1=(N^2-1)/12 w1=R/N m2=(R+1+N)/2 v2=((N-R)^2-1)/12 w2=1-(R/N) mmix=w1m1+w2m2 vmix=w1(v1+m1^2)+w2(v2+m2^2)-mmix^2 Insert algebra. The mean comes out to be (RN+N+N^2-R^2)/2N, matching what you found. The variance comes out to be gross, but just what you get by plugging things in. So we can calculate it pretty quickly for any given N and R. (I plopped those formulas right into R, and added taking a square root). And the standard deviations are not what I was expecting! [ATTACH type="full" alt="1660978038309.png"]258483[/ATTACH] It looks like when you get to rerolling the lowest half of the possible values that the variances start going up again, and can be larger than not dropping anything! This seemed odd, so I calculated them out by getting the distributions in the d4 case, and they do look to be right. [ATTACH type="full" alt="1660977615776.png"]258482[/ATTACH] Anyway, here are what I get for the ratios of the standard deviations for dropping 1s (and keeping the reroll) to the standard deviations of just keeping, for d3 to d100. [ATTACH type="full" alt="1660978609330.png"]258484[/ATTACH] [/QUOTE]
Insert quotes…
Verification
Post reply
Community
General Tabletop Discussion
*Dungeons & Dragons
First playtest thread! One D&D Character Origins.
Top