Question

50. Checking independence For each of the following situations, would you expect the random variables $X$and $Y$to be independent? Explain your answers.

(a)$X$is the rainfall (in inches) on November $6$of this year, and $Y$is the rainfall at the same location on November $6$ of next year.
(b) $X$is the amount of rainfall today, and $Y$ is the rainfall at the same location tomorrow.
(c) $X$is today's rainfall at the airport in Orlando, Florida, and $Y$ is today's rainfall at Disney World just outside Orlando.

Short answer

(a) The amount of rainfall today $X$and, the rainfall at the same location of next year$Y$are not independent.

(b) The amount of rainfall today $X$, and the rainfall at the same location tomorrow $Y$are not independent.

(c) The rainfall at the airport $X$and rainfall at Disney World $Y$are not independent.

Step-by-step solution

Part (a) Step 1: Given information 

Here, $X$is the rainfall (in inches) on November $6$of this year, and $Y$is the rainfall at the same location on November $6$of next year.

Part (a) Step 2: Explanation 

Rainfall on November $6^{th}$this year and the following year at the same spot.

Because the occurrences $X$and $Y$are linked to the particular location, they cannot be presumed to be independent. As a result, the two events are not independent

Part (b) Step 1: Given information 

Let $X$is the amount of rainfall today, and $Y$is the rainfall at the same location tomorrow.

Part (b) Step 2: Explanation 

Today and tomorrow, rain will fall at the identical place.

Because the occurrences $X$and $Y$are linked to a certain area, they cannot be considered independent. As a result, the two scenarios are not independent

Part (c) Step 1: Given information 

Let $X$is today's rainfall at the airport in Orlando, Florida, and $Y$ is today's rainfall at Disney World just outside Orlando.

Part (c) Step 2: Explanation 

Today there was rain in the Orlando airport, as well as at Disneyland.

Because they are linked to specific locations, the events $X$and $Y$cannot be considered as separate. As a result, the two occurrences aren't independent.