Sure, let's analyze each table and determine whether they represent valid probability distributions.
### Table 1
[tex]\[
\begin{array}{|l|l|}
\hline
X & P(X) \\
\hline
1 & 0.42 \\
\hline
2 & 0.18 \\
\hline
5 & 0.34 \\
\hline
7 & 0.06 \\
\hline
\end{array}
\][/tex]
To be a valid probability distribution, the probabilities must satisfy two conditions:
1. Each probability [tex]\( P(X) \)[/tex] must be non-negative.
2. The sum of all probabilities [tex]\( \sum P(X) \)[/tex] must equal 1.
Checking Table 1:
1. Non-negativity: All the probabilities [tex]\( 0.42, 0.18, 0.34, 0.06 \)[/tex] are non-negative.
2. Sum:
[tex]\[
0.42 + 0.18 + 0.34 + 0.06 = 1.00
\][/tex]
Since both conditions are satisfied, Table 1 represents a valid probability distribution.
### Table 2
[tex]\[
\begin{array}{|c|c|}
\hline
X & P(X) \\
\hline
55 & -0.3 \\
\hline
65 & 0.8 \\
\hline
75 & 0.4 \\
\hline
85 & 0.2 \\
\hline
\end{array}
\][/tex]
Checking Table 2:
1. Non-negativity: One probability [tex]\( -0.3 \)[/tex] is negative.
2. Sum:
[tex]\[
-0.3 + 0.8 + 0.4 + 0.2 = 1.1
\][/tex]
Since the sum does not equal 1 and there is a negative probability, Table 2 does not represent a valid probability distribution.
### Table 3
[tex]\[
\begin{array}{|c|c|}
\hline
X & P(X) \\
\hline
-8 & 0.2 \\
\hline
-13 & 0.2 \\
\hline
-15 & 0.2 \\
\hline
-18 & 0.2 \\
\hline
\end{array}
\][/tex]
Checking Table 3:
1. Non-negativity: All probabilities [tex]\( 0.2, 0.2, 0.2, 0.2 \)[/tex] are non-negative.
2. Sum:
[tex]\[
0.2 + 0.2 + 0.2 + 0.2 = 0.8
\][/tex]
Since the sum does not equal 1, Table 3 does not represent a valid probability distribution.
### Summary
- Table 1: Valid probability distribution.
- Table 2: Not a valid probability distribution (negative probability and sum not equal to 1).
- Table 3: Not a valid probability distribution (sum not equal to 1).
Thus, the results are: [True, False, False].
