Question

determine whether each table represents a probability distribution.

x	p(x)
13	0.19
14	0.27
15	0.03
16	0.36

no
yes

x	p(x)
70	0.43
75	0.63
80	0.7
85	0.43
90	0.63

no
yes

x	p(x)
58	0.12
59	0.16
60	0.17
61	-0.02
62	0.14

no
yes

Explanation:

Step1: Recall probability - distribution rules

A probability distribution must satisfy two conditions: 1. \(0\leq P(x)\leq1\) for all \(x\), 2. \(\sum_{x}P(x) = 1\).

Step2: Check the first table

For \(x = 12\), \(P(12)=0.15\); \(x = 13\), \(P(13)=0.19\); \(x = 14\), \(P(14)=0.27\); \(x = 15\), \(P(15)=0.03\); \(x = 16\), \(P(16)=0.36\). All \(P(x)\) values are between \(0\) and \(1\), and \(\sum_{i = 1}^{5}P(x_i)=0.15 + 0.19+0.27 + 0.03+0.36=1\). So it is a probability - distribution.

Step3: Check the second table

\(0.59+0.43 + 0.63+0.7+0.43+0.63=3.41
eq1\). Since the sum of probabilities is not equal to \(1\), it is not a probability - distribution.

Step4: Check the third table

The value \(P(61)=- 0.02\lt0\). Since probabilities must be non - negative, it is not a probability - distribution.

Answer:

1. yes
2. no
3. no