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The PMF (Probability Mass Function) of \( X \) provided in the table can be rewritten mathematically as:

\[
P(X = x) =
\begin{cases}
0.1, & \text{if } x = 1, \\
0.3, & \text{if } x = 2, \\
0.2, & \text{if } x = 3, \\
0.4, & \text{if } x = 4, \\
0, & \text{otherwise.}
\end{cases}
\]

This expresses the probabilities for each value of \( X \), with \( P(X = x) = 0 \) for any \( x \) not explicitly listed.

Let me know if you'd like further clarification! 

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### Relative Questions:
1. How is the PMF \( P(X = x) \) related to the cumulative distribution function (CDF)?
2. What are the properties that a valid PMF must satisfy?
3. Can this PMF represent a continuous random variable? Why or why not?
4. How can we verify that this is a valid PMF mathematically?
5. What is the expected value \( \mathbb{E}[X] \) for this PMF?

### Tip:
Always check that the probabilities in a PMF sum up to 1 to ensure its validity!

Explore the rewritten PMF (Probability Mass Function) for a discrete random variable X. This solution provides a step-by-step approach to representing the given PMF mathematically, ensuring it adheres to the rules of probability. Suitable for Grades 9-12 students.

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