To determine whether the given table represents a valid probability distribution, we need to verify two important properties:
1. Each individual probability [tex]\( P(x) \)[/tex] must lie within the range [tex]\( 0 \leq P(x) \leq 1 \)[/tex].
2. The sum of all probabilities [tex]\( P(x) \)[/tex] should equal 1.
Let's analyze the given data:
[tex]\[
\begin{tabular}{|c|c|}
\hline \multicolumn{2}{|c}{\begin{tabular}{c}
Probability \\
Distribution $A$
\end{tabular}} \\
\hline$X$ & $P(x)$ \\
\hline 0 & 0 \\
\hline 1 & 0 \\
\hline 2 & 0 \\
\hline
\end{tabular}
\][/tex]
Step 1: Verify that each probability lies within the range [0, 1]:
- For [tex]\( P(0) = 0 \)[/tex], it lies within the range [0, 1].
- For [tex]\( P(1) = 0 \)[/tex], it lies within the range [0, 1].
- For [tex]\( P(2) = 0 \)[/tex], it lies within the range [0, 1].
Since all the individual probabilities are within the valid range, we move to the next step.
Step 2: Verify that the sum of all probabilities equals 1:
Calculate the sum of the probabilities:
[tex]\[
P(0) + P(1) + P(2) = 0 + 0 + 0 = 0
\][/tex]
After performing the summation, we see that the total probability is [tex]\( 0 \)[/tex].
Conclusion:
For a valid probability distribution, the sum of the probabilities must be 1. However, in this case, the sum of the probabilities is [tex]\( 0 \)[/tex], which does not equal 1. Therefore, the given table does not represent a valid probability distribution.
