Introduction:
In this question, we are given two variables X and Y. X represents the amount of rainfall in a randomly selected month in our hometown, and Y represents the number of children in Los Angeles during the same month. We need to determine whether X and Y are independent or not.

Definition of Independence:
To understand whether X and Y are independent, we need to know what it means for two variables to be independent. Two variables are said to be independent if the occurrence or value of one does not affect the occurrence or value of the other.

Explanation:
To determine if X and Y are independent, we need to consider the relationship between rainfall in our hometown and the number of children in Los Angeles. Let's analyze this relationship in detail:

1. Rainfall in our hometown:
- The amount of rainfall in our hometown is determined by weather patterns, atmospheric conditions, and other local factors.
- It is not influenced by the number of children in Los Angeles or any other external factors.
- Rainfall is a natural occurrence and does not depend on human activities like population.

2. Number of children in Los Angeles:
- The number of children in Los Angeles is influenced by factors such as birth rates, immigration, emigration, and other social and economic factors.
- The number of children is not influenced by the amount of rainfall in our hometown or any other external weather conditions.

Conclusion:
Based on the above analysis, we can conclude that the amount of rainfall in our hometown (X) and the number of children in Los Angeles (Y) are independent variables. The occurrence or value of one variable does not affect the occurrence or value of the other. The variables X and Y are influenced by different factors and are not related to each other. Therefore, X and Y are independent.