- A valid probability distribution requires all probabilities to be between \(0\) and \(1\) and their sum to be exactly \(1\).
- Table 1 sums to \(0.95\), failing the sum condition.
- Table 2 sums to \(1.0\) and all probabilities are valid, meeting both requirements.
- Tables 3 and 4 fail because some values are outside the range \([0, 1]\).
- The valid table is \(\boxed{\text{Table 2}}\).
Explanation
- Analyze the criteria for a valid probability distribution
A probability distribution for a discrete random variable \(X\) must adhere to two fundamental properties:
- Each individual probability \(P(X_i)\) must fall within the range of \(0\) and \(1\) inclusive (i.e., \(0 \leq P(X_i) \leq 1\)).
- The sum of all probabilities in the distribution must exactly equal \(1\) (i.e., \(\sum P(X_i) = 1\)).
We will evaluate each table based on these criteria to determine which one represents a valid probability distribution.
- Evaluate each table against the criteria
- Table 1: Probabilities are \(\{0.2, 0.35, 0.15, 0.25\}\).
- All values are between 0 and 1.
- The sum is \(0.2 + 0.35 + 0.15 + 0.25 = 0.95\). Since \(0.95
eq 1\), this is not a valid distribution.
- Table 2: Probabilities are \(\{0.4, 0.15, 0.25, 0.2\}\).
- All values are between 0 and 1.
- The sum is \(0.4 + 0.15 + 0.25 + 0.2 = 1.0\). This satisfies both conditions.
- Table 3: Probabilities are \(\{0.4, 1.05, 0.02, 0.35\}\).
- Here, \(1.05 > 1\), which violates the first condition. Also, the sum is \(1.82
eq 1\). This is not valid.
- Table 4: Probabilities are \(\{-0.2, -0.1, 0.4, 0.3\}\).
- Here, \(-0.2\) and \(-0.1\) are less than 0, violating the first condition. The sum is also \(0.4
eq 1\). This is not valid.
- Conclusion and final answer
After checking all four tables, only Table 2 satisfies both requirements: every probability value is between 0 and 1, and the sum of all probabilities is exactly 1. Therefore, Table 2 is the only valid probability distribution. The final answer is \(\boxed{\text{Table 2}}\).
Examples
In data analysis, checking if a set of probabilities forms a valid distribution is a common prerequisite before calculating expected values or variances. For example, if you are modeling the likelihood of different outcomes in a manufacturing process, ensuring your probability model sums to 1 is essential to ensure your model's reliability in predicting future defects or performance metrics.