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Sagot :
To determine which of the given probability distributions represents a valid probability distribution, we need to evaluate each distribution against two fundamental requirements:
1. All probabilities must be between 0 and 1 inclusive.
2. The sum of all probabilities must be equal to 1.
Let's examine each distribution step by step:
Probability Distribution A:
[tex]\[ \begin{array}{|c|c|} \hline X & P(x) \\ \hline 1 & -0.14 \\ \hline 2 & 0.6 \\ \hline 3 & 0.25 \\ \hline 4 & 0.29 \\ \hline \end{array} \][/tex]
1. Check if all [tex]\( P(x) \)[/tex] values are between 0 and 1:
- [tex]\( -0.14 \)[/tex] is not between 0 and 1 (Invalid).
- [tex]\( 0.6 \)[/tex] is between 0 and 1.
- [tex]\( 0.25 \)[/tex] is between 0 and 1.
- [tex]\( 0.29 \)[/tex] is between 0 and 1.
Given that [tex]\(-0.14\)[/tex] is not valid, Distribution A cannot be a probability distribution. However, for completeness, let's check the sum:
2. Sum of probabilities:
- [tex]\( -0.14 + 0.6 + 0.25 + 0.29 = 1.0 \)[/tex]
- Although the sum is 1, the presence of a negative probability makes it invalid.
Probability Distribution B:
[tex]\[ \begin{array}{|c|c|} \hline X & P(x) \\ \hline 1 & 0 \\ \hline 2 & 0.45 \\ \hline 3 & 0.16 \\ \hline 4 & 0.39 \\ \hline \end{array} \][/tex]
1. Check if all [tex]\( P(x) \)[/tex] values are between 0 and 1:
- [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.
All values are valid probabilities.
2. Sum of probabilities:
- [tex]\( 0 + 0.45 + 0.16 + 0.39 = 1.0 \)[/tex].
Distribution B satisfies both conditions and is a valid probability distribution.
Probability Distribution C:
[tex]\[ \begin{array}{|c|c|} \hline X & P(x) \\ \hline 1 & 0.45 \\ \hline 2 & 1.23 \\ \hline 3 & -0.87 \\ \hline \end{array} \][/tex]
1. Check if all [tex]\( P(x) \)[/tex] values are between 0 and 1:
- [tex]\( 0.45 \)[/tex] is between 0 and 1.
- [tex]\( 1.23 \)[/tex] is greater than 1 (Invalid).
- [tex]\( -0.87 \)[/tex] is less than 0 (Invalid).
Given that [tex]\( 1.23 \)[/tex] and [tex]\( -0.87 \)[/tex] are out of the valid range, Distribution C is invalid. Nevertheless, check the sum for completeness:
2. Sum of probabilities:
- [tex]\( 0.45 + 1.23 + (-0.87) = 0.81 \neq 1 \)[/tex].
Distribution C does not sum to 1 and has invalid probabilities.
Conclusion:
Among the given distributions, only Probability Distribution B is a valid probability distribution.
1. All probabilities must be between 0 and 1 inclusive.
2. The sum of all probabilities must be equal to 1.
Let's examine each distribution step by step:
Probability Distribution A:
[tex]\[ \begin{array}{|c|c|} \hline X & P(x) \\ \hline 1 & -0.14 \\ \hline 2 & 0.6 \\ \hline 3 & 0.25 \\ \hline 4 & 0.29 \\ \hline \end{array} \][/tex]
1. Check if all [tex]\( P(x) \)[/tex] values are between 0 and 1:
- [tex]\( -0.14 \)[/tex] is not between 0 and 1 (Invalid).
- [tex]\( 0.6 \)[/tex] is between 0 and 1.
- [tex]\( 0.25 \)[/tex] is between 0 and 1.
- [tex]\( 0.29 \)[/tex] is between 0 and 1.
Given that [tex]\(-0.14\)[/tex] is not valid, Distribution A cannot be a probability distribution. However, for completeness, let's check the sum:
2. Sum of probabilities:
- [tex]\( -0.14 + 0.6 + 0.25 + 0.29 = 1.0 \)[/tex]
- Although the sum is 1, the presence of a negative probability makes it invalid.
Probability Distribution B:
[tex]\[ \begin{array}{|c|c|} \hline X & P(x) \\ \hline 1 & 0 \\ \hline 2 & 0.45 \\ \hline 3 & 0.16 \\ \hline 4 & 0.39 \\ \hline \end{array} \][/tex]
1. Check if all [tex]\( P(x) \)[/tex] values are between 0 and 1:
- [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.
All values are valid probabilities.
2. Sum of probabilities:
- [tex]\( 0 + 0.45 + 0.16 + 0.39 = 1.0 \)[/tex].
Distribution B satisfies both conditions and is a valid probability distribution.
Probability Distribution C:
[tex]\[ \begin{array}{|c|c|} \hline X & P(x) \\ \hline 1 & 0.45 \\ \hline 2 & 1.23 \\ \hline 3 & -0.87 \\ \hline \end{array} \][/tex]
1. Check if all [tex]\( P(x) \)[/tex] values are between 0 and 1:
- [tex]\( 0.45 \)[/tex] is between 0 and 1.
- [tex]\( 1.23 \)[/tex] is greater than 1 (Invalid).
- [tex]\( -0.87 \)[/tex] is less than 0 (Invalid).
Given that [tex]\( 1.23 \)[/tex] and [tex]\( -0.87 \)[/tex] are out of the valid range, Distribution C is invalid. Nevertheless, check the sum for completeness:
2. Sum of probabilities:
- [tex]\( 0.45 + 1.23 + (-0.87) = 0.81 \neq 1 \)[/tex].
Distribution C does not sum to 1 and has invalid probabilities.
Conclusion:
Among the given distributions, only Probability Distribution B is a valid probability distribution.
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