Question

the following table shows the probability distribution for the number of children in each family in the town of crossville. what is the average number of children in this town?
x | 0 | 1 | 2 | 3 | 4 | 5
p(x) | 0.3 | 0.1 | 0.3 | 0.1 | 0.1 | 0.1
7.
options: 2.1, 0.3, 1.9, 3
clear all

Explanation:

Step1: Recall the formula for the expected value (average) of a discrete probability distribution.

The formula for the expected value \( E(X) \) (which is the average in this context) of a discrete random variable \( X \) is \( E(X)=\sum_{i} x_i \cdot P(x_i) \), where \( x_i \) are the possible values of \( X \) and \( P(x_i) \) are their corresponding probabilities.

Step2: Calculate each term \( x_i \cdot P(x_i) \)

• For \( x = 0 \): \( 0\times0.3 = 0 \)
• For \( x = 1 \): \( 1\times0.1 = 0.1 \)
• For \( x = 2 \): \( 2\times0.3 = 0.6 \)
• For \( x = 3 \): \( 3\times0.1 = 0.3 \)
• For \( x = 4 \): \( 4\times0.1 = 0.4 \)
• For \( x = 5 \): \( 5\times0.1 = 0.5 \)

Step3: Sum up all the terms

Now, we sum these products: \( 0 + 0.1 + 0.6 + 0.3 + 0.4 + 0.5 \)
First, \( 0+0.1 = 0.1 \)
Then, \( 0.1 + 0.6 = 0.7 \)
Then, \( 0.7 + 0.3 = 1.0 \)
Then, \( 1.0 + 0.4 = 1.4 \)
Then, \( 1.4 + 0.5 = 1.9 \)

Answer:

1.9