Probability Mass Function
The probability distribution of a discrete random variable is represented by its probability mass function. It is a function whose domain contains the set of discrete values that the random variable can assume, with the probabilities of the random variable assuming the values in the domain as its range. The probability mass function of the discrete random variable Y is denoted by pY(y), and is defined as
pY(y) = P { Y = y }, y is an element of the domain of pY(y)
0 £ pY(y) £ 1 for all values of y that Y can assume
pY(y) = 0 for all other values of y
A probability mass function is often depicted graphically by a probability histogram. The potential values of the random variable are plotted on the x-axis, while their associated probabilities are plotted on the y-axis. Rectangles of equal width are centered on each discrete value, and their heights are equal to the probabilities that the random variable can assume those values.
EX. A random variable X assumes a value equal to the sum of the rolls of two dice. Since there are 6 possible values for each die, the fundamental principle of counting asserts that there are 6 × 6 = 36 possible outcomes of the dice roll. With unbiased dice, each outcome will have a 1/36 chance of occurring.
The following table shows the relationship between the outcome of the dice roll and the value of the random variable X:
| Value of X | Dice Roll (x , y) x - roll of first die y - roll of second die |
Number of Outcomes |
| Â | Â | Â |
| 2 | (1,1) | 1 |
| 3 | (1,2) , (2,1) | 2 |
| 4 | (1,3) , (2,2) , (3,1) | 3 |
| 5 | (1,4) , (2,3) , (3,2) , (4,1) | 4 |
| 6 | (1,5) , (2,4) , (3,3) , (4,2) , (5,1) | 5 |
| 7 | (1,6) , (2,5) , (3,4) , (4,3) , (5,2) , (6,1) | 6 |
| 8 | (2,6) , (3,5) , (4,4) , (5,3) , (6,2) | 5 |
| 9 | (3,6) , (4,5) , (5,4) , (6,3) | 4 |
| 10 | (4,6) , (5,5) , (6,4) | 3 |
| 11 | (5,6) , (6,5) | 2 |
| 12 | (6,6) | 1 |
| Â | Â | Â |
| Total | Â | 36 |
Therefore, the probability mass function of X is
| x | px(x) |
| Â | Â |
| 2 | 1 / 36 |
| 3 | 2/ 36 |
| 4 | 3/ 36 |
| 5 | 4/ 36 |
| 6 | 5/ 36 |
| 7 | 6/ 36 |
| 8 | 5/ 36 |
| 9 | 4/ 36 |
| 10 | 3/ 36 |
| 11 | 2/ 36 |
| 12 | 1/ 36 |
Note that all the probabilities sum up to 1, i.e.
The probability histogram for the random variable X is graphed as follows:
Â
