Independent and identically distributed random variables (IID)

Instance of: Property of a set of random variables

AKA: i.i.d., iid, or IID, random sample

Distinct from:

English: Independent means the value drawn for one doesn’t influence which value will be drawn from another, or alternatively you don’t learn anything about the value of one from the value of another. Identically distributed means that the chances of a value for one variable are the same for the other, if one is drawn from a normal(0,1) then the other is drawn from a normal(0,1). Substantively it’s ruling out trends, where say the distribution’s mean or variance drifts as you take samples from it, e.g. with a time series.

Formalization:

Two random variables \(X\) and \(Y\) are identically distributed if and only if \[ F_x(x)= F_y(x) \forall x \in I \] ::: {.column-margin} Where \(F\) is the cumulative probability distribution. :::

And independent if and only if \[ F_{x,y}(x,y)=F_x(x) * F_y(x) \forall x,y \in I \]

Cites: Wikipedia) ; Wikidata

Code

Imports and spin up toy data objects and databases.

toy_vector_numeric <- c(1,2,3,4,5)
toy_vector_character <- c('a','b','c','d','e')
toy_matrix <- matrix(1:9, nrow=3,ncol=3)
toy_list <- list('a','1',T,c('red','green'))
toy_df <- data.frame(id=c('unit1','unit2','unit3'), y=c(1,2,3), x= c(3,2,1))

toy_dirty_df <- data.frame(id=c('','NA','NaN','inf'), y=c(1,2,3,4), x= c(0,NA,NaN,Inf)) #can't explicitly include NULL 

library(data.table)
toy_dt <- data.table(id=c('unit1','unit2','unit3'), y=c(1,2,3), x= c(3,2,1))
toy_dirty_dt <- as.data.table(toy_dirty_df)
library(tidyverse)

── Attaching packages ─────────────────────────────────────── tidyverse 1.3.2 ──
✔ ggplot2 3.4.0      ✔ purrr   0.3.5 
✔ tibble  3.1.8      ✔ dplyr   1.0.10
✔ tidyr   1.2.1      ✔ stringr 1.5.0 
✔ readr   2.1.3      ✔ forcats 0.5.2 
── Conflicts ────────────────────────────────────────── tidyverse_conflicts() ──
✖ dplyr::between()   masks data.table::between()
✖ dplyr::filter()    masks stats::filter()
✖ dplyr::first()     masks data.table::first()
✖ dplyr::lag()       masks stats::lag()
✖ dplyr::last()      masks data.table::last()
✖ purrr::transpose() masks data.table::transpose()

library(arrow)


Attaching package: 'arrow'

The following object is masked from 'package:utils':

    timestamp

import numpy as np
toy_vector_numeric = np.array([1,2,3,4,5])
toy_vector_character = np.array(['a','b','c','d','e'])
toy_list = ['a','1',True,['red','green']]
toy_dictionary = { 'a':1 , 'b':2, 'c':3}

from jax import numpy as jnp
toy_vector_numeric_jax = jnp.array([1,2,3,4,5])
#toy_vector_character_jax = jnp.array(['a','b','c','d','e']) #only numeric is allowed in jax

WARNING:jax._src.lib.xla_bridge:No GPU/TPU found, falling back to CPU. (Set TF_CPP_MIN_LOG_LEVEL=0 and rerun for more info.)

import pandas as pd
toy_df = pd.DataFrame(data={'id': ['unit1','unit2','unit3'], 'y': [1, 2, 3], 'x': [3, 2, 1]})

import torch

import tensorflow as tf

import pyarrow as pa

library(DBI)
# Create an ephemeral in-memory RSQLite database
#con <- dbConnect(RSQLite::SQLite(), dbname = ":memory:")
#dbListTables(con)
#dbWriteTable(con, "mtcars", mtcars)
#dbListTables(con)

#Configuration failed because libpq was not found. Try installing:
#* deb: libpq-dev libssl-dev (Debian, Ubuntu, etc)
#install.packages('RPostgres')
#remotes::install_github("r-dbi/RPostgres")
#Took forever because my file permissions were broken
#pg_lsclusters
require(RPostgres)
# Connect to the default postgres database
#I had to follow these instructions and create both a username and database that matched my ubuntu name
#https://www.digitalocean.com/community/tutorials/how-to-install-postgresql-on-ubuntu-20-04-quickstart
con_Postgres <- dbConnect(RPostgres::Postgres())


DROP TABLE IF EXISTS toy_df;


CREATE TABLE IF NOT EXISTS toy_df (
  id varchar(5),
    y INTEGER,
    x INTEGER
);


INSERT INTO toy_df (id, y, x)
VALUES
    ('unit1',1,3),
    ('unit2',2,2),
    ('unit3',3,1);

#install.packages("duckdb")
library("DBI")
con_duckdb = dbConnect(duckdb::duckdb(), ":memory:")

#pip install duckdb==0.6.0
import duckdb
con_duckdb = duckdb.connect()