Certainly! Let's go through the steps to determine if Probability Distribution A is a valid probability distribution.
A valid probability distribution must satisfy the following two conditions:
1. The probability of each event [tex]\( P(x) \)[/tex] must be between 0 and 1 inclusive.
2. The sum of the probabilities of all the events must be equal to 1.
Let's evaluate Probability Distribution A:
[tex]\[
\begin{tabular}{|c|c|}
\hline \multicolumn{2}{|c}{Probability Distribution A} \\
\hline X & P(x) \\
\hline 0 & 0 \\
\hline 1 & 0 \\
\hline 2 & 0 \\
\hline 3 & 1 \\
\hline
\end{tabular}
\][/tex]
Step-by-Step Validation:
1. Check Each Probability Value:
- [tex]\( P(0) = 0 \)[/tex] is between 0 and 1.
- [tex]\( P(1) = 0 \)[/tex] is between 0 and 1.
- [tex]\( P(2) = 0 \)[/tex] is between 0 and 1.
- [tex]\( P(3) = 1 \)[/tex] is between 0 and 1.
All values of [tex]\( P(x) \)[/tex] are within the required range.
2. Sum of All Probabilities:
Let's find the sum of these probabilities:
[tex]\[
\sum P(x) = P(0) + P(1) + P(2) + P(3) = 0 + 0 + 0 + 1 = 1
\][/tex]
The sum of the probabilities is 1.
Since both conditions are satisfied, we can conclude that:
Probability Distribution A is a valid probability distribution.
