SOLUTION: An internet service provider charges its customers for the time of the internet use rounding it up to the nearest hour. The joint distribution of the used time (X, hours) and the c

Question 1198231: An internet service provider charges its customers for the time of the internet use rounding it up to the nearest hour. The joint distribution of the used time (X, hours) and the charge per hour (Y , cents) is given in the table below.
___________________x___________
P(x, y)___1_____2_____3____4___
_____1____0___0.06__0.06__0.10
__y__2__0.10__0.10__0.04__0.04
_____3__0.40__0.10___0_____0
Each customer is charged Z = X · Y cents, which is the number of hours multiplied by the price of each hour. Find the distribution of Z.
Answer by CPhill(1959) (Show Source): You can put this solution on YOUR website!
**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).

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