List the possible values for each of the variables. For example, if X were the value rolled on a regular six-sided die and Y were the value rolled on a second six-sided die, you would have two lists of numbers from one to six. Because there is the same chance of landing on any side of a die, the probabilities for X and Y have a uniform distribution. Graphs of the probabilities would show straight lines going from one to six and having a height of 1/6 (0.16) for each value of the die.
Calculate the possible values for the sum of X and Y. In the dice example, the lowest values for each die would be one, so the lowest value for the sum would be 1 + 1 = 2. The highest value for the sum would 6 + 6 = 12. The other values for the sum would be the integers between one and 12.
For each possible value of the sum, list the permutations of X and Y that would yield that sum. The number of permutations for each value shows the distribution for the sum.
Multiply the number of possible values for X by the number of possible values for Y, to get the total number of permutations. In the example using two dice, the total number of permutations is 6 * 6 = 36.
Divide the number of permutations for each possible value of the sum by 36. The results of these divisions yield the probabilities for each possible sum. This distribution is the convolution of the two variables. The probability of getting a sum of two is 1/36, or 0.028, which is also the probability of getting a sum of 12. The probability of getting a seven, which is the mean of the sums, is 6/36, or 0.16.