Question 1198231
**1. Calculate the Probability of Each Z Value**

* **Z = 1:** 
    * P(Z=1) = P(X=1, Y=1) = 0.06

* **Z = 2:** 
    * P(Z=2) = P(X=1, Y=2) + P(X=2, Y=1) = 0.10 + 0.10 = 0.20

* **Z = 3:** 
    * P(Z=3) = P(X=1, Y=3) + P(X=3, Y=1) = 0.06 + 0.40 = 0.46

* **Z = 4:** 
    * P(Z=4) = P(X=2, Y=2) + P(X=4, Y=1) = 0.10 + 0.10 = 0.20

* **Z = 6:** 
    * P(Z=6) = P(X=2, Y=3) + P(X=3, Y=2) = 0.04 + 0.04 = 0.08

* **Z = 12:** 
    * P(Z=12) = P(X=4, Y=3) = 0.10

**2. Summarize the Distribution of Z**

| Z | 1 | 2 | 3 | 4 | 6 | 12 |
|---|---|---|---|---|---|---|
| P(Z) | 0.06 | 0.20 | 0.46 | 0.20 | 0.08 | 0.10 |

**This table represents the probability distribution of the total charge (Z) for the internet service provider.**

**Key Points:**

* The distribution of Z is discrete, as Z can only take on certain integer values.
* The probabilities for each value of Z are calculated by summing the probabilities of all combinations of X and Y that result in that value of Z.

This analysis provides the probability distribution of the total charge (Z) for the given joint probability distribution of X (used time) and Y (charge per hour).