To find the attendant's expected earnings for this time period, we'll need to calculate the expected value of the function $g(X) = 2X - 1$. The expected value of $g(X)$ is given by:

$E[g(X)] = \sum_{x} g(x) \cdot P(X = x)$

Given:

• $X$ represents the number of cars.
• The probability distribution for $X$ is:

$P(X = 4) = \frac{1}{12}, \quad P(X = 5) = \frac{1}{12}, \quad P(X = 6) = \frac{1}{4}, \quad P(X = 7) = \frac{1}{4}, \quad P(X = 8) = \frac{1}{6}, \quad P(X = 9) = \frac{1}{6}$

• The function $g(x) = 2x - 1$ represents the amount of money paid.

Now, let's calculate $g(x)$ for each value of $x$:

$g(4) = 2(4) - 1 = 8 - 1 = 7$ $g(5) = 2(5) - 1 = 10 - 1 = 9$ $g(6) = 2(6) - 1 = 12 - 1 = 11$ $g(7) = 2(7) - 1 = 14 - 1 = 13$ $g(8) = 2(8) - 1 = 16 - 1 = 15$ $g(9) = 2(9) - 1 = 18 - 1 = 17$

Next, multiply each value of $g(x)$ by the corresponding probability $P(X = x)$ and sum the results to find the expected value $E[g(X)]$:

$E[g(X)] = 7 \times \frac{1}{12} + 9 \times \frac{1}{12} + 11 \times \frac{1}{4} + 13 \times \frac{1}{4} + 15 \times \frac{1}{6} + 17 \times \frac{1}{6}$

Calculate each term:

$7 \times \frac{1}{12} = \frac{7}{12}$ $9 \times \frac{1}{12} = \frac{9}{12}$ $11 \times \frac{1}{4} = \frac{11}{4} = \frac{33}{12}$ $13 \times \frac{1}{4} = \frac{13}{4} = \frac{39}{12}$ $15 \times \frac{1}{6} = \frac{15}{6} = \frac{30}{12}$ $17 \times \frac{1}{6} = \frac{17}{6} = \frac{34}{12}$

Now sum these values:

$E[g(X)] = \frac{7}{12} + \frac{9}{12} + \frac{33}{12} + \frac{39}{12} + \frac{30}{12} + \frac{34}{12} = \frac{152}{12} = 12.67 \text{ dollars}$

Thus, the expected earnings for the attendant during this time period are approximately $12.67.

Would you like more details or have any other questions?

Here are 5 related questions:

1. How do you calculate the expected value for a different function of $X$?
2. What if the probability distribution of $X$ changes? How would that affect the expected earnings?
3. Can you explain the concept of expected value in general?
4. How would you calculate the variance of the earnings $g(X)$?
5. What is the significance of the function $g(x)$ in practical applications?

Tip: When calculating expected value, always make sure to multiply the function values by their corresponding probabilities and sum them up.