At Westonci.ca, we make it easy for you to get the answers you need from a community of knowledgeable individuals. Our Q&A platform offers a seamless experience for finding reliable answers from experts in various disciplines. Explore comprehensive solutions to your questions from a wide range of professionals on our user-friendly 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.
Thank you for choosing our platform. We're dedicated to providing the best answers for all your questions. Visit us again. Thank you for your visit. We're dedicated to helping you find the information you need, whenever you need it. Keep exploring Westonci.ca for more insightful answers to your questions. We're here to help.
