Question

consider the roll of a pair of dice, and let X denote the sum of the two numbers appearing on the dice. find the probability distribution of X and find the cumulative distribution function. what is the most likely sum

Answer 1

The first step is calculating the sample space S= {2, 3, ,3 ,4,4,4 ,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,8,8,8,8,8,9,9,9,9,10,10,10,11,11,12}

count the frequency of each possible outcome and divide with total count n =36 to find PDF then calculate CDF for each X by adding the current and all preceding PDFs.

X PDF CDF

2 $\frac 1 {36}$ $\frac 1 {36}$

3 $\frac 1 {18}$ $\frac {1} {12}$

4 $\frac 1 {12}$ $\frac 16$

5 $\frac 19$ $\frac 5 {18}$

6 $\frac 5 {36}$ $\frac 5 {12}$

7 $\frac 1 6$ $\frac 7 {12}$

8 $\frac 5 {36}$ $\frac {13} {18}$

9 $\frac 1 9$ $\frac 5 6$

10 $\frac 1 {12}$ $\frac {11} {12}$

11 $\frac 1 {18}$ $\frac {35} {36}$

12 $\frac 1 {36}$ 1

The most likely sum is 7 it has the highest PDF .