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A probability distribution of a random variable X is a function that gives the probability of each outcome in the sample space. It can be represented by tables, equations, or graphs. A probability distribution needs to satisfy two conditions to be valid.
A probability distribution can be either discrete like the geometric distribution and the binomial distribution, or continuous like the normal distribution.
If a probability distribution is based on mathematical models and assumptions, then it is called a theoretical probability distribution. On the other hand, an experimental probability distribution is determined by conducting an experiment.
Consider the roll of a pair of standard dice. Let X be the random variable that represents the sum of the two dice. By the fundamental counting principle, since rolling each die has 6 possible outcomes, there are a total of 6* 6 = 36 possible results. Additionally, the possible values of X are integers from 2 to 12.
A table that represents the theoretical probability distribution of X will now be created. Frequencies represent the number of dice roll results that add up to the given values x of the random variable X. The frequency is divided by 36 to determine the theoretical probability of each outcome.
| X=Sum of Two Dice | ||
|---|---|---|
| x | Frequency | P(X=x) |
| 2 | 1 | 1/36≈ 0.028 |
| 3 | 2 | 2/36≈ 0.056 |
| 4 | 3 | 3/36≈0.083 |
| 5 | 4 | 4/36≈0.111 |
| 6 | 5 | 5/36≈0.139 |
| 7 | 6 | 6/36≈0.167 |
| 8 | 5 | 5/36≈0.139 |
| 9 | 4 | 4/36≈0.111 |
| 10 | 3 | 3/36≈0.083 |
| 11 | 2 | 2/36≈ 0.056 |
| 12 | 1 | 1/36≈ 0.028 |
The probability distribution of X can also be represented by a bar graph. The possible outcomes of X are marked on the horizontal axis and their probabilities are presented on the vertical axis.
In this example, each possible value of the random variable can be associated with its corresponding probability because it is a discrete random variable. This is also why the bars of the probability distribution must be separated.