Final answer:
The random variable X, representing the face value of a roll of a fair six-sided die, can take on one of six possible values: 1, 2, 3, 4, 5, or 6. Each value has an equal probability of occurring, and the distribution of X in this context is uniform.
Step-by-step explanation:
A fair die is a cube with six faces, each face having a distinct number of dots ranging from 1 up to 6. When you roll the die, the top face shows a random number, which is the outcome of your roll. If X is a random variable representing the face value on the throw of a fair die, the possible values of X are the integer values 1, 2, 3, 4, 5, and 6. Each of these values has an equal chance of occurring due to the die's fairness, so they constitute the sample space or the set of all possible outcomes of the experiment.
In various contexts, such as games or probability experiments, this random variable can be defined in different ways:
- In a game where the number of dice showing a one is of interest, X could be defined as the number of ones observed after rolling several dice.
- If money is won or lost based on the outcome of the roll, X might represent the monetary gain or loss.
- For an event such as rolling a number at least five, the random variable would be concerned with the occurrence of that event, which includes just the outcomes 5 and 6.
The distribution of a random variable like X, when a die is rolled, is uniform, as each outcome has an equal probability of 1/6.