Question

question 1? which table is a probability distribution table? o x 1 2 3 4 p 0.2 0.35 0.15 0.25 o x 1 2 3 4 p 0.4 1.05 0.02 0.35 o x 1 2 3 4 p -0.2 -0.1 0.4 0.3 o x 1 2 3 4 p 0.4 0.15 0.25 0.2

Explanation:

Step1: Recall probability - distribution rules

A probability - distribution table must satisfy two conditions: 1. Each probability \(P(x)\) must be between \(0\) and \(1\) (inclusive), i.e., \(0\leq P(x)\leq1\). 2. The sum of all probabilities \(\sum_{i}P(x_{i}) = 1\).

Step2: Check the first table

For the first table: \(P(1)=0.2\), \(P(2)=0.35\), \(P(3)=0.15\), \(P(4)=0.25\). \(\sum_{i = 1}^{4}P(i)=0.2 + 0.35+0.15 + 0.25=0.95
eq1\), so it is not a probability - distribution table.

Step3: Check the second table

For the second table, \(P(2)=1.05>1\), which violates the condition that \(0\leq P(x)\leq1\) for all \(x\), so it is not a probability - distribution table.

Step4: Check the third table

For the third table, \(P(1)= - 0.2<0\) and \(P(2)=-0.1<0\), which violates the condition that \(0\leq P(x)\leq1\) for all \(x\), so it is not a probability - distribution table.

Step5: Check the fourth table

For the fourth table: \(P(1)=0.4\), \(P(2)=0.15\), \(P(3)=0.25\), \(P(4)=0.2\). \(\sum_{i = 1}^{4}P(i)=0.4 + 0.15+0.25 + 0.2=1\), and \(0\leq P(i)\leq1\) for \(i = 1,2,3,4\).

Answer:

The fourth table (where \(x = 1,2,3,4\) and \(P(1)=0.4\), \(P(2)=0.15\), \(P(3)=0.25\), \(P(4)=0.2\)) is a probability - distribution table.