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To determine which of the given options represents a valid probability distribution, we need to verify that two conditions are met for each distribution:
1. All probabilities must be between 0 and 1: That is, for each [tex]\(P(x)\)[/tex], [tex]\(0 \leq P(x) \leq 1\)[/tex].
2. The sum of all probabilities must be equal to 1: That is, [tex]\(\sum P(x) = 1\)[/tex].
Let's analyze each probability distribution in detail:
### Probability Distribution A:
[tex]\[ \begin{tabular}{|c|c|} \hline $X$ & $P(x)$ \\ \hline 1 & -0.14 \\ \hline 2 & 0.6 \\ \hline 3 & 0.25 \\ \hline 4 & 0.29 \\ \hline \end{tabular} \][/tex]
- Checking if all probabilities are between 0 and 1:
- [tex]\(P(1) = -0.14\)[/tex] (Not between 0 and 1)
- [tex]\(P(2) = 0.6\)[/tex] (Between 0 and 1)
- [tex]\(P(3) = 0.25\)[/tex] (Between 0 and 1)
- [tex]\(P(4) = 0.29\)[/tex] (Between 0 and 1)
Since [tex]\(P(1)\)[/tex] is [tex]\(-0.14\)[/tex] (which is not between 0 and 1), Probability Distribution A is not valid.
### Probability Distribution B:
[tex]\[ \begin{tabular}{|c|c|} \hline $X$ & $P( x )$ \\ \hline 1 & 0 \\ \hline 2 & 0.45 \\ \hline 3 & 0.16 \\ \hline 4 & 0.39 \\ \hline \end{tabular} \][/tex]
- Checking if all probabilities are between 0 and 1:
- [tex]\(P(1) = 0\)[/tex] (Between 0 and 1)
- [tex]\(P(2) = 0.45\)[/tex] (Between 0 and 1)
- [tex]\(P(3) = 0.16\)[/tex] (Between 0 and 1)
- [tex]\(P(4) = 0.39\)[/tex] (Between 0 and 1)
- Checking if the sum of all probabilities is 1:
- [tex]\(P(1) + P(2) + P(3) + P(4) = 0 + 0.45 + 0.16 + 0.39 = 1\)[/tex]
Since all probabilities are between 0 and 1 and their sum is 1, Probability Distribution B is valid.
### Probability Distribution C:
[tex]\[ \begin{tabular}{|c|c|} \hline $X$ & $P(x)$ \\ \hline 1 & 0.45 \\ \hline 2 & 1.23 \\ \hline 3 & -0.87 \\ \hline \end{tabular} \][/tex]
- Checking if all probabilities are between 0 and 1:
- [tex]\(P(1) = 0.45\)[/tex] (Between 0 and 1)
- [tex]\(P(2) = 1.23\)[/tex] (Not between 0 and 1)
- [tex]\(P(3) = -0.87\)[/tex] (Not between 0 and 1)
Since [tex]\(P(2) = 1.23\)[/tex] and [tex]\(P(3) = -0.87\)[/tex] which are not between 0 and 1, Probability Distribution C is not valid.
### Conclusion:
After analyzing all three distributions, only Probability Distribution B represents a valid probability distribution.
1. All probabilities must be between 0 and 1: That is, for each [tex]\(P(x)\)[/tex], [tex]\(0 \leq P(x) \leq 1\)[/tex].
2. The sum of all probabilities must be equal to 1: That is, [tex]\(\sum P(x) = 1\)[/tex].
Let's analyze each probability distribution in detail:
### Probability Distribution A:
[tex]\[ \begin{tabular}{|c|c|} \hline $X$ & $P(x)$ \\ \hline 1 & -0.14 \\ \hline 2 & 0.6 \\ \hline 3 & 0.25 \\ \hline 4 & 0.29 \\ \hline \end{tabular} \][/tex]
- Checking if all probabilities are between 0 and 1:
- [tex]\(P(1) = -0.14\)[/tex] (Not between 0 and 1)
- [tex]\(P(2) = 0.6\)[/tex] (Between 0 and 1)
- [tex]\(P(3) = 0.25\)[/tex] (Between 0 and 1)
- [tex]\(P(4) = 0.29\)[/tex] (Between 0 and 1)
Since [tex]\(P(1)\)[/tex] is [tex]\(-0.14\)[/tex] (which is not between 0 and 1), Probability Distribution A is not valid.
### Probability Distribution B:
[tex]\[ \begin{tabular}{|c|c|} \hline $X$ & $P( x )$ \\ \hline 1 & 0 \\ \hline 2 & 0.45 \\ \hline 3 & 0.16 \\ \hline 4 & 0.39 \\ \hline \end{tabular} \][/tex]
- Checking if all probabilities are between 0 and 1:
- [tex]\(P(1) = 0\)[/tex] (Between 0 and 1)
- [tex]\(P(2) = 0.45\)[/tex] (Between 0 and 1)
- [tex]\(P(3) = 0.16\)[/tex] (Between 0 and 1)
- [tex]\(P(4) = 0.39\)[/tex] (Between 0 and 1)
- Checking if the sum of all probabilities is 1:
- [tex]\(P(1) + P(2) + P(3) + P(4) = 0 + 0.45 + 0.16 + 0.39 = 1\)[/tex]
Since all probabilities are between 0 and 1 and their sum is 1, Probability Distribution B is valid.
### Probability Distribution C:
[tex]\[ \begin{tabular}{|c|c|} \hline $X$ & $P(x)$ \\ \hline 1 & 0.45 \\ \hline 2 & 1.23 \\ \hline 3 & -0.87 \\ \hline \end{tabular} \][/tex]
- Checking if all probabilities are between 0 and 1:
- [tex]\(P(1) = 0.45\)[/tex] (Between 0 and 1)
- [tex]\(P(2) = 1.23\)[/tex] (Not between 0 and 1)
- [tex]\(P(3) = -0.87\)[/tex] (Not between 0 and 1)
Since [tex]\(P(2) = 1.23\)[/tex] and [tex]\(P(3) = -0.87\)[/tex] which are not between 0 and 1, Probability Distribution C is not valid.
### Conclusion:
After analyzing all three distributions, only Probability Distribution B represents a valid probability distribution.
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