Westonci.ca is the best place to get answers to your questions, provided by a community of experienced and knowledgeable experts. Explore thousands of questions and answers from a knowledgeable community of experts ready to help you find solutions. Get precise and detailed answers to your questions from a knowledgeable community of experts on our Q&A platform.
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.
Thanks for using our platform. We're always here to provide accurate and up-to-date answers to all your queries. We hope our answers were useful. Return anytime for more information and answers to any other questions you have. Keep exploring Westonci.ca for more insightful answers to your questions. We're here to help.