To determine which of the provided probability distributions is valid, we need to verify that the sum of the probabilities for each distribution equals 1. This property is essential for any valid probability distribution.
Let's check each distribution step-by-step.
### Probability Distribution A:
[tex]\[
\begin{array}{|c|c|}
\hline
X & P(X) \\
\hline
1 & 0.42 \\
\hline
2 & 0.38 \\
\hline
3 & 0.13 \\
\hline
4 & 0.07 \\
\hline
\end{array}
\][/tex]
Sum of probabilities for Distribution A:
[tex]\[
0.42 + 0.38 + 0.13 + 0.07 = 1.0
\][/tex]
The sum is 1.0, so Distribution A is a valid probability distribution.
### Probability Distribution B:
[tex]\[
\begin{array}{|c|c|}
\hline
X & P(X) \\
\hline
1 & 0.27 \\
\hline
2 & 0.28 \\
\hline
3 & 0.26 \\
\hline
4 & 0.27 \\
\hline
\end{array}
\][/tex]
Sum of probabilities for Distribution B:
[tex]\[
0.27 + 0.28 + 0.26 + 0.27 = 1.08
\][/tex]
The sum is 1.08, which is not equal to 1. Therefore, Distribution B is not a valid probability distribution.
### Probability Distribution C:
[tex]\[
\begin{array}{|c|c|}
\hline
X & P(X) \\
\hline
1 & 0.16 \\
\hline
2 & 0.39 \\
\hline
\end{array}
\][/tex]
Sum of probabilities for Distribution C:
[tex]\[
0.16 + 0.39 = 0.55
\][/tex]
The sum is 0.55, which is not equal to 1. Therefore, Distribution C is not a valid probability distribution.
### Conclusion:
Among the given probability distributions, only Probability Distribution A is a valid distribution because its probabilities sum to 1.
