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Which of the following represents a valid probability distribution?

\begin{tabular}{|c|c|}
\hline \multicolumn{2}{|c|}{ Probability Distribution A} \\
\hline [tex]$X$[/tex] & [tex]$P(x)$[/tex] \\
\hline 1 & -0.14 \\
\hline 2 & 0.6 \\
\hline 3 & 0.25 \\
\hline 4 & 0.29 \\
\hline
\end{tabular}

\begin{tabular}{|c|c|}
\hline \multicolumn{2}{|c|}{ Probability Distribution B} \\
\hline [tex]$X$[/tex] & [tex]$P(x)$[/tex] \\
\hline 1 & 0 \\
\hline 2 & 0.45 \\
\hline 3 & 0.16 \\
\hline 4 & 0.39 \\
\hline
\end{tabular}

\begin{tabular}{|c|c|}
\hline \multicolumn{2}{|c|}{ Probability Distribution C} \\
\hline [tex]$X$[/tex] & [tex]$P(x)$[/tex] \\
\hline 1 & 0.45 \\
\hline 2 & 1.23 \\
\hline 3 & -0.87 \\
\hline
\end{tabular}

Sagot :

GinnyAnsweridn
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.
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