Question

determine whether each table represents a probability distribution.

x	p(x)
13	0.27
14	0.3
15	0.06
16	0.04

o yes
o no

x	p(x)
45	0.5
50	0.68
55	0.53
60	0.54
65	0.65

o no
o yes

x	p(x)
86	0.31
87	0.11
88	0.18
89	-0.02
90	0.29

o no
o yes

Explanation:

Step1: Recall probability - distribution rules

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

Step2: Check the first table

For \(x = 12\), \(P(12)=0.33\); \(x = 13\), \(P(13)=0.27\); \(x = 14\), \(P(14)=0.3\); \(x = 15\), \(P(15)=0.06\); \(x = 16\), \(P(16)=0.04\).
\(\sum_{x = 12}^{16}P(x)=0.33 + 0.27+0.3 + 0.06+0.04=1\) and \(0\leq P(x)\leq1\) for all \(x\). So it is a probability - distribution.

Step3: Check the second table

\(P(40)=0.59\), \(P(45)=0.5\), \(P(50)=0.68\), \(P(55)=0.53\), \(P(60)=0.54\), \(P(65)=0.65\).
\(\sum_{x = 40}^{65}P(x)=0.59 + 0.5+0.68+0.53+0.54+0.65 = 3.59>1\). So it is not a probability - distribution.

Step4: Check the third table

\(P(89)=- 0.02<0\). Since probabilities must be non - negative, it is not a probability - distribution.

Answer:

yes
no
no