A Pair of Dice and Random Variables
Consider an experiment in which we toss a
pair of dice. We can at the most basic level observe the resulting pair of face
values.
|
1st d6Þ ß 2nd d6 |
1 |
2 |
3 |
4 |
5 |
6 |
|
1 |
(1,1) |
(2,1) |
(3,1) |
(4,1) |
(5,1) |
(6,1) |
|
2 |
(1,2) |
(2,2) |
(3,2) |
(4,2) |
(5,2) |
(6,2) |
|
3 |
(1,3) |
(2,3) |
(3,3) |
(4,3) |
(5,3) |
(6,3) |
|
4 |
(1,4) |
(2,4) |
(3,4) |
(4,4) |
(5,4) |
(6,4) |
|
5 |
(1,5) |
(2,5) |
(3,5) |
(4,5) |
(5,5) |
(6,5) |
|
6 |
(1,6) |
(2,6) |
(3,6) |
(4,6) |
(5,6) |
(6,6) |
In this example the dice are assumed to be
six-sided dice, each with face values {1,2,3,4,5,6}.
We consider several examples of random
variable definitions for this experiment. We then compute the random variable one
pair at a time. Each pair yields a value for the random variable.
SUM = (1st d6) + (2nd
d6)
|
1st d6Þ ß 2nd d6 [sum] |
1 |
2 |
3 |
4 |
5 |
6 |
|
1 |
(1,1) [2] |
(2,1) [3] |
(3,1) [4] |
(4,1) [5] |
(5,1) [6] |
(6,1) [7] |
|
2 |
(1,2) [3] |
(2,2) [4] |
(3,2) [5] |
(4,2) [6] |
(5,2) [7] |
(6,2) [8] |
|
3 |
(1,3) [4] |
(2,3) [5] |
(3,3) [6] |
(4,3) [7] |
(5,3) [8] |
(6,3) [9] |
|
4 |
(1,4) [5] |
(2,4) [6] |
(3,4) [7] |
(4,4) [8] |
(5,4) [9] |
(6,4) [10] |
|
5 |
(1,5) [6] |
(2,5) [7] |
(3,5) [8] |
(4,5) [9] |
(5,5) [10] |
(6,5) [11] |
|
6 |
(1,6) [7] |
(2,6) [8] |
(3,6) [9] |
(4,6) [10] |
(5,6) [11] |
(6,6) [12] |
Values for SUM: 2,3,4,5,6,7,8,9,10,11,12
Pairs to SUM:
(1,1) →
[2]
(2,1), (1,2)
→ [3]
(3,1), (2,2),
(1,3) → [4]
(4,1), (2,3),
(3,2), (4,1) → [5]
(5,1), (4,2),
(3,3), (2,4), (1,5) → [6]
(6,1), (5,2),
(4,3), (3,4), (2,5), (1,6) → [7]
(6,2), (5,3),
(4,4), (3,5), (2,6) → [8]
(6,3), (5,4),
(4,5), (3,6) → [9]
(6,4), (5,5),
(4,6) → [10]
(6,5), (5,6)
→ [11]
(6,6) →
[12]
PRODUCT = (1st d6)*(2nd
d6)
|
1st d6Þ ß 2nd d6 [product] |
1 |
2 |
3 |
4 |
5 |
6 |
|
1 |
(1,1) [1] |
(2,1) [2] |
(3,1) [3] |
(4,1) [4] |
(5,1) [5] |
(6,1) [6] |
|
2 |
(1,2) [2] |
(2,2) [4] |
(3,2) [6] |
(4,2) [8] |
(5,2) [10] |
(6,2) [12] |
|
3 |
(1,3) [3] |
(2,3) [6] |
(3,3) [9] |
(4,3) [12] |
(5,3) [15] |
(6,3) [18] |
|
4 |
(1,4) [4] |
(2,4) [8] |
(3,4) [12] |
(4,4) [16] |
(5,4) [20] |
(6,4) [24] |
|
5 |
(1,5) [5] |
(2,5) [10] |
(3,5) [15] |
(4,5) [20] |
(5,5) [25] |
(6,5) [30] |
|
6 |
(1,6) [6] |
(2,6) [12] |
(3,6) [18] |
(4,6) [24] |
(5,6) [30] |
(6,6) [36] |
Values for PRODUCT:
1,2,3,4,5,6,8,9,10,12,15,16,18,20,24,25,30,36
Pairs to PRODUCT:
(1,1) →
[1]
(2,1), (1,2)
→ [2]
(3,1), (1,3)
→ [3]
(4,1), (2,2),
(1,4) → [4]
(5,1), (1,5)
→ [5]
(6,1), (3,2),
(2,3), (1,6) → [6]
(4,2), (2,4)
→ [8]
(3,3)
→ [9]
(5,2), (2,5)
→ [10]
(6,2), (4,3),
(3,4), (2,6) → [12]
(5,3), (3,5)
→ [15]
(4,4) →
[16]
(6,3), (3,6)
→ [18]
(5,4), (4,5)
→ [20]
(6,4), (4,6)
→ [24]
(5,5) →
[25]
(6,5), (5,6)
→ [30]
(6,6) →
[36]
HIGHTIE = higher of two face values or
common value if tied
|
1st d6Þ ß 2nd d6 [hightie] |
1 |
2 |
3 |
4 |
5 |
6 |
|
1 |
(1,1) [1] |
(2,1) [2] |
(3,1) [3] |
(4,1) [4] |
(5,1) [5] |
(6,1) [6] |
|
2 |
(1,2) [2] |
(2,2) [2] |
(3,2) [3] |
(4,2) [4] |
(5,2) [5] |
(6,2) [6] |
|
3 |
(1,3) [3] |
(2,3) [3] |
(3,3) [3] |
(4,3) [4] |
(5,3) [5] |
(6,3) [6] |
|
4 |
(1,4) [4] |
(2,4) [4] |
(3,4) [4] |
(4,4) [4] |
(5,4) [5] |
(6,4) [6] |
|
5 |
(1,5) [5] |
(2,5) [5] |
(3,5) [5] |
(4,5) [5] |
(5,5) [5] |
(6,5) [6] |
|
6 |
(1,6) [6] |
(2,6) [6] |
(3,6) [6] |
(4,6) [6] |
(5,6) [6] |
(6,6) [6] |
Values for HIGHTIE: 1,2,3,4,5,6
Pairs to HIGHTIE:
(1,1) →
[1]
(2,1), (1,2)
→ [2]
(3,1), (3,2),
(2,3), (1,3) → [3]
(4,1), (4,3),
(4,2), (4,1), (1,4), (2,4), (3,4), (4,4) → [4]
(5,1), (5,2),
(5,3), (5,4), (5,5), (4,5), (3,5), (2,5), (1,5) → [5]
(6,1), (6,2),
(6,3), (6,4), (6,5), (6,6), (5,6), (4,6), (3,6), (2,6), (1,6) → [6]
LOWTIE = lower of two face values or
common value if tied
|
1st d6Þ ß 2nd d6 [low] |
1 |
2 |
3 |
4 |
5 |
6 |
|
1 |
(1,1) [1] |
(2,1) [1] |
(3,1) [1] |
(4,1) [1] |
(5,1) [1] |
(6,1) [1] |
|
2 |
(1,2) [1] |
(2,2) [2] |
(3,2) [3] |
(4,2) [2] |
(5,2) [2] |
(6,2) [2] |
|
3 |
(1,3) [1] |
(2,3) [2] |
(3,3) [3] |
(4,3) [3] |
(5,3) [3] |
(6,3) [3] |
|
4 |
(1,4) [1] |
(2,4) [2] |
(3,4) [3] |
(4,4) [4] |
(5,4) [4] |
(6,4) [4] |
|
5 |
(1,5) [1] |
(2,5) [2] |
(3,5) [3] |
(4,5) [4] |
(5,5) [5] |
(6,5) [5] |
|
6 |
(1,6) [1] |
(2,6) [2] |
(3,6) [3] |
(4,6) [4] |
(5,6) [5] |
(6,6) [6] |
Values for LOWTIE: 1,2,3,4,5,6
Pairs to LOWTIE:
(1,1), (2,1),
(3,1), (4,1), (5,1), (6,1), (1,6), (1,5), (1,4), (1,3), (1,2), (1,1) →
[1]
(2,2), (3,2),
(4,2), (5,2), (6,2), (2,6), (2,5), (2,4), (2,3) → [2]
(3,3), (4,3),
(5,3), (6,3), (3,6), (3,5), (3,4) → [3]
(4,4), (5,4),
(6,4), (5,4) → [4]
(5,5), (6,5),
(5,6) → [5]
(6,6) →
[6]
EVENODD = 1 if 1st d6 even and
2nd d6 odd, 0 otherwise
|
1st d6Þ ß 2nd d6 [evenodd] |
1 |
2 |
3 |
4 |
5 |
6 |
|
1 |
(1,1) [0] |
(2,1) [1] |
(3,1) [0] |
(4,1) [1] |
(5,1) [0] |
(6,1) [1] |
|
2 |
(1,2) [0] |
(2,2) [0] |
(3,2) [0] |
(4,2) [0] |
(5,2) [0] |
(6,2) [0] |
|
3 |
(1,3) [0] |
(2,3) [1] |
(3,3) [0] |
(4,3) [1] |
(5,3) [0] |
(6,3) [1] |
|
4 |
(1,4) [0] |
(2,4) [0] |
(3,4) [0] |
(4,4) [0] |
(5,4) [0] |
(6,4) [0] |
|
5 |
(1,5) [0] |
(2,5) [1] |
(3,5) [0] |
(4,5) [1] |
(5,5) [0] |
(6,5) [1] |
|
6 |
(1,6) [0] |
(2,6) [0] |
(3,6) [0] |
(4,6) [0] |
(5,6) [0] |
(6,6) [0] |
Values for EVENODD: 1,0
Pairs to EVENODD:
(2,1), (2,3), (2,5), (4,1), (4,3), (4,5), (6,1), (6,3),
(6,5) → [1]
(2,2), (2,4), (2,6), (4,2), (4,4), (4,6), (6,2), (6,4),
(6,6),
(1,1), (1,3), (1,5), (3,1), (3,3), (3,5), (5,1), (5,3),
(5,5),
(1,2), (1,4), (1,6), (3,2), (3,4), (3,6), (5,2), (5,4),
(5,6), (1,1) → [0]
ODDEVEN = 1 if 1st d6 odd and 2nd
d6 even, 0 otherwise
|
1st d6Þ ß 2nd d6 [oddeven] |
1 |
2 |
3 |
4 |
5 |
6 |
|
1 |
(1,1) [0] |
(2,1) [0] |
(3,1) [0] |
(4,1) [0] |
(5,1) [0] |
(6,1) [0] |
|
2 |
(1,2) [1] |
(2,2) [0] |
(3,2) [1] |
(4,2) [0] |
(5,2) [1] |
(6,2) [0] |
|
3 |
(1,3) [0] |
(2,3) [0] |
(3,3) [0] |
(4,3) [0] |
(5,3) [0] |
(6,3) [0] |
|
4 |
(1,4) [1] |
(2,4) [0] |
(3,4) [1] |
(4,4) [0] |
(5,4) [1] |
(6,4) [0] |
|
5 |
(1,5) [0] |
(2,5) [0] |
(3,5) [0] |
(4,5) [0] |
(5,5) [0] |
(6,5) [0] |
|
6 |
(1,6) [1] |
(2,6) [0] |
(3,6) [1] |
(4,6) [0] |
(5,6) [1] |
(6,6) [0] |
Values for ODDEVEN: 1,0
Pairs to ODDEVEN:
(1,2), (1,4), (1,6), (3,2), (3,4), (3,6), (5,2), (5,4),
(5,6), (1,1) → [1]
(2,2), (2,4), (2,6), (4,2), (4,4), (4,6), (6,2), (6,4),
(6,6),
(1,1), (1,3), (1,5), (3,1), (3,3), (3,5), (5,1), (5,3),
(5,5),
(2,1), (2,3), (2,5), (4,1), (4,3), (4,5), (6,1), (6,3),
(6,5) → [0]
EVENEVEN = 1 if both faces even, 0
otherwise
|
1st d6Þ ß 2nd d6 [eveneven] |
1 |
2 |
3 |
4 |
5 |
6 |
|
1 |
(1,1) [0] |
(2,1) [0] |
(3,1) [0] |
(4,1) [0] |
(5,1) [0] |
(6,1) [0] |
|
2 |
(1,2) [0] |
(2,2) [1] |
(3,2) [0] |
(4,2) [1] |
(5,2) [0] |
(6,2) [1] |
|
3 |
(1,3) [0] |
(2,3) [0] |
(3,3) [0] |
(4,3) [0] |
(5,3) [0] |
(6,3) [0] |
|
4 |
(1,4) [0] |
(2,4) [1] |
(3,4) [0] |
(4,4) [1] |
(5,4) [0] |
(6,4) [1] |
|
5 |
(1,5) [0] |
(2,5) [0] |
(3,5) [0] |
(4,5) [0] |
(5,5) [0] |
(6,5) [0] |
|
6 |
(1,6) [0] |
(2,6) [1] |
(3,6) [0] |
(4,6) [1] |
(5,6) [0] |
(6,6) [1] |
Values for EVENEVEN: 1,0
Pairs to EVENEVEN:
(2,2), (2,4), (2,6), (4,2), (4,4), (4,6), (6,2), (6,4),
(6,6) → [1]
(1,2), (1,4), (1,6), (3,2), (3,4), (3,6), (5,2), (5,4),
(5,6),
(1,1), (1,3), (1,5), (3,1), (3,3), (3,5), (5,1), (5,3),
(5,5),
(2,1), (2,3), (2,5), (4,1), (4,3), (4,5), (6,1), (6,3),
(6,5) → [0]
ODDODD = 1 if both faces odd, 0 otherwise
|
1st d6Þ ß 2nd d6 [oddodd] |
1 |
2 |
3 |
4 |
5 |
6 |
|
1 |
(1,1) [1] |
(2,1) [0] |
(3,1) [1] |
(4,1) [0] |
(5,1) [1] |
(6,1) [0] |
|
2 |
(1,2) [0] |
(2,2) [0] |
(3,2) [0] |
(4,2) [0] |
(5,2) [0] |
(6,2) [0] |
|
3 |
(1,3) [1] |
(2,3) [0] |
(3,3) [1] |
(4,3) [0] |
(5,3) [1] |
(6,3) [0] |
|
4 |
(1,4) [0] |
(2,4) [0] |
(3,4) [0] |
(4,4) [0] |
(5,4) [0] |
(6,4) [0] |
|
5 |
(1,5) [1] |
(2,5) [0] |
(3,5) [1] |
(4,5) [0] |
(5,5) [1] |
(6,5) [0] |
|
6 |
(1,6) [0] |
(2,6) [0] |
(3,6) [0] |
(4,6) [0] |
(5,6) [0] |
(6,6) [0] |
Values for ODDODD: 1,0
Pairs to ODDODD:
(1,1), (1,3), (1,5), (3,1), (3,3), (3,5), (5,1), (5,3),
(5,5) → [1]
(1,2), (1,4), (1,6), (3,2), (3,4), (3,6), (5,2), (5,4),
(5,6),
(2,2), (2,4), (2,6), (4,2), (4,4), (4,6), (6,2), (6,4),
(6,6),
(2,1), (2,3), (2,5), (4,1), (4,3), (4,5), (6,1), (6,3),
(6,5) → [0]