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A probability distribution is a mathematical function that provides the probabilities of occurrence of different possible outcomes in an experiment.
In the context of our problem, each outcome of the function \(R(x)\) corresponds to a value \(x\) from 0 to 4. We are provided with a probability \(R(x) = 0.2\) for each of these values:

• \(R(0) = 0.2\)
• \(R(1) = 0.2\)
• \(R(2) = 0.2\)
• \(R(3) = 0.2\)
• \(R(4) = 0.2\)

The sum of all these probabilities must equal 1 for \(R(x)\) to be considered a valid probability distribution.
This requirement ensures that the function accounts for all possible outcomes. Let's verify this:\[0.2 + 0.2 + 0.2 + 0.2 + 0.2 = 1\]Since the total is 1, we can confirm that \(R(x)\) is a valid probability distribution. Every value of \(x\) has an equal chance of occurring, making this a uniform distribution.

Creating a histogram is an excellent way to visually represent a probability distribution. A histogram displays the frequency of data represented as bars where the height of each bar corresponds to the probability of each outcome. In this exercise,

• Each value \(x = 0, 1, 2, 3, 4\) corresponds to a separate bar.
• The height of each bar is determined by the probability value, \(0.2\) in this instance.
• All bars must be of equal height since each value has the same probability.

Although it’s challenging to draw graphs or create visual diagrams in a text-based medium, imagine each of these bars as consistent in height, represented evenly side by side.
This visualization method helps in quickly understanding how probabilities are distributed, especially when examining how uniform or varied the distribution is.

Verifying a probability function is crucial to ensure that it meets the criteria of a valid probability distribution. There are a few simple checks required to verify this:

• The sum of all probabilities must be 1.
• Each individual probability must be at least 0 and at most 1.

In this exercise, with \(R(x) = 0.2\) for each \(x\) from 0 to 4, we confirm that all probabilities are within the acceptable range:

• Checked: 0.2 is between 0 and 1.
• Total sum: 0.2 + 0.2 + 0.2 + 0.2 + 0.2 = 1.

Both of these conditions are satisfied, affirming the legitimacy of \(R(x)\) as a probability function. This verification ensures that the function correctly represents a real-world probabilistic scenario, ensuring all possible outcomes are accounted for.