Answered

Welcome to Westonci.ca, where your questions are met with accurate answers from a community of experts and enthusiasts. Join our Q&A platform to connect with experts dedicated to providing precise answers to your questions in different areas. Connect with a community of professionals ready to help you find accurate solutions to your questions quickly and efficiently.

Which of the following represents a valid probability distribution?

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

Sagot :

GinnyAnswer
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.
Thanks for stopping by. We strive to provide the best answers for all your questions. See you again soon. Thank you for your visit. We're dedicated to helping you find the information you need, whenever you need it. We're here to help at Westonci.ca. Keep visiting for the best answers to your questions.