Answer :
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].
### 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].