To determine which of the given probability distributions are valid, we need to verify if the sum of probabilities in each distribution equals 1. A valid probability distribution must have probabilities that sum to exactly 1.
Let's analyze each distribution in detail:
### Probability Distribution A:
Given probabilities:
[tex]\[P(X = 1) = 0.42\][/tex]
[tex]\[P(X = 2) = 0.38\][/tex]
[tex]\[P(X = 3) = 0.13\][/tex]
[tex]\[P(X = 4) = 0.07\][/tex]
Sum of probabilities:
[tex]\[0.42 + 0.38 + 0.13 + 0.07 = 1.0\][/tex]
Since the sum is 1.0, Distribution A is a valid probability distribution.
### Probability Distribution B:
Given probabilities:
[tex]\[P(X = 1) = 0.27\][/tex]
[tex]\[P(X = 2) = 0.28\][/tex]
[tex]\[P(X = 3) = 0.26\][/tex]
[tex]\[P(X = 4) = 0.27\][/tex]
Sum of probabilities:
[tex]\[0.27 + 0.28 + 0.26 + 0.27 = 1.08\][/tex]
Since the sum is 1.08, which is not equal to 1, Distribution B is not a valid probability distribution.
### Probability Distribution C:
Given probabilities:
[tex]\[P(X = 1) = 0.16\][/tex]
[tex]\[P(X = 2) = 0.39\][/tex]
[tex]\[P(X = 3) = 0.18\][/tex]
Sum of probabilities:
[tex]\[0.16 + 0.39 + 0.18 = 0.73\][/tex]
Since the sum is 0.73, which is not equal to 1, Distribution C is not a valid probability distribution.
### Summary:
- Probability Distribution A: Valid (sum = 1.0)
- Probability Distribution B: Not Valid (sum = 1.08)
- Probability Distribution C: Not Valid (sum = 0.73)
Thus, the only valid probability distribution among the given options is Probability Distribution A.
