DATA605: Assignment #6

6.1.6

A die is rolled twice. Let X denote the sum of the two numbers that turn up, and Y the difference of the numbers (specifically, the number on the first roll minus the number on the second). Show that E(XY ) = E(X)E(Y ). Are X and Y independent?

library( dplyr )
#make a dataframe to hold each possible roll
die1 <- c( rep( 1,6 ), rep( 2,6 ), rep( 3,6 ), rep( 4,6 ), rep( 5,6 ), rep( 6,6 ))
die2 <- c( rep( 1:6,6 ) )
combo_df <- data.frame( 'roll1' = die1, 'roll2' = die2 )
combo_df <- combo_df %>% mutate( X = roll1 + roll2, Y = abs(roll1 - roll2), XY = X*Y )
head( combo_df )

##   roll1 roll2 X Y XY
## 1     1     1 2 0  0
## 2     1     2 3 1  3
## 3     1     3 4 2  8
## 4     1     4 5 3 15
## 5     1     5 6 4 24
## 6     1     6 7 5 35

find the mean of the columns…

combo_means <- colMeans( combo_df )
combo_means

##     roll1     roll2         X         Y        XY 
##  3.500000  3.500000  7.000000  1.944444 13.611111

Does mean( XY ) == mean(X)*mean(Y)?

combo_means['XY'] == combo_means['X']*combo_means['Y']

##   XY 
## TRUE

This shows that, yes, E(XY ) = E(X)E(Y )

The code below simulates this process and finds the values X, Y and XY

#simulate the double dice roll 'n' times
n <- 10000
roll1 <- sample(1:6, n, replace=TRUE)
roll2 <- sample(1:6, n, replace=TRUE)
#use dplyr methods
die_df <- data.frame( 'roll1' = roll1, 'roll2' = roll2 )
die_df <- die_df %>% mutate( X = roll1 + roll2, Y = abs(roll1 - roll2), XY = X*Y )
head( die_df )

##   roll1 roll2  X Y XY
## 1     4     6 10 2 20
## 2     1     5  6 4 24
## 3     1     3  4 2  8
## 4     3     2  5 1  5
## 5     3     3  6 0  0
## 6     2     1  3 1  3

Now find the mean of the columns…

diemeans <- colMeans( die_df )
diemeans

##   roll1   roll2       X       Y      XY 
##  3.4648  3.5096  6.9744  1.9468 13.5922

Does mean( XY ) == mean(X)*mean(Y)?

diemeans['XY']

##      XY 
## 13.5922

diemeans['X']*diemeans['Y']

##        X 
## 13.57776

…those simulated values look pretty close