To determine which of the provided probability distributions is valid, we must check each distribution against the criteria for being a valid probability distribution:
1. Each probability value must be between 0 and 1 (inclusive).
2. The sum of the probabilities must be exactly 1.
### Probability Distribution A
[tex]\[
\begin{array}{|c|c|}
\hline \multicolumn{2}{|c|}{ Probability Distribution A } \\
\hline X & P(x) \\
\hline 1 & -0.14 \\
\hline 2 & 0.6 \\
\hline 3 & 0.25 \\
\hline 4 & 0.29 \\
\hline
\end{array}
\][/tex]
1. Check the range of each probability:
- [tex]\( -0.14 \)[/tex] is not between 0 and 1, so it fails the first criterion.
Therefore, Probability Distribution A is not valid.
### Probability Distribution B
[tex]\[
\begin{array}{|c|c|}
\hline \multicolumn{2}{|c|}{ Probability Distribution B } \\
\hline X & P(x) \\
\hline 1 & 0 \\
\hline 2 & 0.45 \\
\hline 3 & 0.16 \\
\hline 4 & 0.39 \\
\hline
\end{array}
\][/tex]
1. Check the range of each probability:
- [tex]\( 0 \)[/tex] is between 0 and 1.
- [tex]\( 0.45 \)[/tex] is between 0 and 1.
- [tex]\( 0.16 \)[/tex] is between 0 and 1.
- [tex]\( 0.39 \)[/tex] is between 0 and 1.
2. Sum the probabilities:
[tex]\[
0 + 0.45 + 0.16 + 0.39 = 1
\][/tex]
Since all probabilities are within the required range and their sum is 1, Probability Distribution B is valid.
### Probability Distribution C
[tex]\[
\begin{array}{|c|c|}
\hline \multicolumn{2}{|c|}{ Probability Distribution C } \\
\hline X & P(x) \\
\hline 1 & 0.45 \\
\hline 2 & 1.23 \\
\hline 3 & -0.87 \\
\hline
\end{array}
\][/tex]
1. Check the range of each probability:
- [tex]\( 0.45 \)[/tex] is between 0 and 1.
- [tex]\( 1.23 \)[/tex] is not between 0 and 1.
- [tex]\( -0.87 \)[/tex] is not between 0 and 1.
Since some probabilities are not within the required range, Probability Distribution C is not valid.
### Conclusion
Only Probability Distribution B represents a valid probability distribution.
