- A valid probability distribution must have all probabilities \(0 \leq P(x_i) \leq 1\) and their total sum \(\sum P(x_i) = 1\).
- Table 1 contains negative values, violating the first condition.
- Table 3 sums to \(0.95\), violating the second condition.
- Table 4 contains a value \(1.05 > 1\), violating the first condition.
- Table 2 satisfies both conditions: \(0.4 + 0.15 + 0.25 + 0.2 = 1\).
- The correct table is the second one: \(\boxed{\text{Table 2}}\).
Explanation
- Understand the conditions for a valid probability distribution
To determine which of the provided tables represents a valid probability distribution, we must verify two fundamental axioms of probability for each discrete random variable \(X\):
- Non-negativity and Boundedness: For every possible outcome \(x_i\), the probability \(P(x_i)\) must be between \(0\) and \(1\) inclusive, i.e., \(0 \leq P(x_i) \leq 1\).
- Normalization: The sum of the probabilities of all possible outcomes must be exactly \(1\), i.e., \(\sum P(x_i) = 1\).
- Evaluate each table according to the probability axioms
- Table 1: The probabilities are \(\{-0.2, -0.1, 0.4, 0.3\}\). Since \(-0.2\) and \(-0.1\) are less than \(0\), this violates the first condition.
- Table 2: The probabilities are \(\{0.4, 0.15, 0.25, 0.2\}\).
- Check condition 1: All values are between \(0\) and \(1\). (Pass)
- Check condition 2: \(0.4 + 0.15 + 0.25 + 0.2 = 1.0\). (Pass)
- Since both conditions are satisfied, this table represents a valid probability distribution.
- Table 3: The probabilities are \(\{0.2, 0.35, 0.15, 0.25\}\).
- Check condition 1: All values are between \(0\) and \(1\). (Pass)
- Check condition 2: \(0.2 + 0.35 + 0.15 + 0.25 = 0.95\). Since \(0.95
eq 1\), this violates the second condition.
- Table 4: The probabilities are \(\{0.4, 1.05, 0.02, 0.35\}\). Since \(1.05 > 1\), this violates the first condition.
- Conclusion of the evaluation
By checking each table, we find that only Table 2 satisfies both the requirement that all probabilities must be between \(0\) and \(1\) and the requirement that the sum of all probabilities must equal \(1\). Therefore, Table 2 is the only valid probability distribution.
Examples
In statistics, verifying probability distributions is crucial for modeling real-world uncertainty, such as analyzing the likelihood of different outcomes in a game, financial risk assessment, or quality control. For instance, if you are designing a survey and assign weights (probabilities) to various demographic responses, you must ensure these weights add up to 1 and are non-negative so that your predictions remain mathematically sound and representative of the total population. If \(P_1 = 0.4\), \(P_2 = 0.15\), \(P_3 = 0.25\), and \(P_4 = 0.2\), then \(\sum P_i = 0.4 + 0.15 + 0.25 + 0.2 = 1\), confirming that the model covers the entire sample space.