You roll two fair 6-sided dice. Let be the sum of the two independent rolls.
(a) Give the PMF of in table form.
(b) Give a mathematical expression for the PMF of .
You roll two fair 6-sided dice. Let X be the sum of the two independent rolls.
(a) Give the PMF of X in table form.
(b) Give a mathematical expression for the PMF of X .
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To find the probability mass function (PMF) of the sum of two independent rolls of fair 6-sided dice, we need to consider all possible outcomes and their respective probabilities.
(a) PMF of the sum in table form:
Let's list all possible combinations of the two dice rolls and calculate their probabilities:
Sum (X) | Outcome | Probability (P(X))
-------------------------------------
2 | 1, 1 | 1/36
3 | 1, 2 | 1/36
| 2, 1 | 1/36
4 | 1, 3 | 1/36
| 2, 2 | 1/36
| 3, 1 | 1/36
5 | 1, 4 | 1/36
| 2, 3 | 1/36
| 3, 2 | 1/36
| 4, 1 | 1/36
6 | 1, 5 | 1/36
| 2, 4 | 1/36
| 3, 3 | 1/36
| 4, 2 | 1/36
| 5, 1 | 1/36
7 | 1, 6 | 1/36
| 2, 5 | 1/36
| 3, 4 | 1/36
| 4, 3 | 1/36
| 5, 2 | 1/36
| 6, 1 | 1/36
8 | 2, 6 | 1/36
| 3, 5 | 1/36
| 4, 4 | 1/36
| 5, 3 | 1/36
| 6, 2 | 1/36
9 | 3, 6 | 1/36
| 4, 5 | 1/36
| 5, 4 | 1/36
| 6, 3 | 1/36
10 | 4, 6 | 1/36
| 5, 5 | 1/36
| 6, 4 | 1/36
11 | 5, 6 | 1/36
| 6, 5 | 1/36
12 | 6, 6 | 1/36
(b) Mathematical expression for the PMF of X:
To generalize the PMF for any sum X, we can express it as:
{ 1/36, if X = 2
P(X) = { 2/36, if X = 3
{ 3/36, if X = 4
{ 4/36, if X = 5
{ 5/36, if X = 6
{ 6/36, if X = 7
{ 5/36, if X = 8
{ 4/36, if X = 9
{ 3/36, if X = 10
{ 2/36, if X = 11
{ 1/36, if X = 12
This expression represents the probability of obtaining each sum X when rolling two fair 6-sided dice independently.