Let's break down the problem.

Part a) Finding the height of the function, $f(x)$

The function $f(x)$ represents a uniform probability distribution. For a uniform distribution, the total area under the function should equal 1 because the total probability must be 1.

From the graph:

• The function has a constant height over the interval $[0, 12]$, which is the base of the rectangle.
• The width of the rectangle is 12 (from $x = 0$ to $x = 12$).

To find the height $f(x)$, we use the fact that the area of the rectangle must equal 1: $\text{Area} = \text{height} \times \text{width}$ Since the total probability (area) is 1: $\text{height} \times 12 = 1$ Solving for the height: $\text{height} = \frac{1}{12}$

Thus, the height of the function $f(x)$ is $\frac{1}{12}$.

Part b) Probability of the shaded region

The shaded region corresponds to the interval $[9, 12]$ on the $x$-axis. The probability of an interval in a uniform distribution is proportional to the area under the function over that interval.

To find this probability, we calculate the area of the shaded rectangle: $\text{Probability} = \text{height} \times \text{width of the shaded region}$ The height is $\frac{1}{12}$, and the width of the shaded region (from 9 to 12) is $12 - 9 = 3$. Thus, the probability is: $\frac{1}{12} \times 3 = \frac{3}{12} = \frac{1}{4}$

So, the probability of the shaded region is $\frac{1}{4}$.

Summary of Answers:

a) The height of the function $f(x)$ is $\frac{1}{12}$.
b) The probability of the shaded region is $\frac{1}{4}$.

Would you like any further details or clarification?

Here are five related questions to further expand on this:

1. How would you find the probability for a different interval, say from $x = 5$ to $x = 10$?
2. What happens to the height of $f(x)$ if the interval changes to $[0, 15]$?
3. Can this uniform distribution model be applied to any other real-world scenarios?
4. How would you interpret the cumulative distribution function (CDF) for this uniform distribution?
5. What is the expected value $E(x)$ for this uniform distribution?

Tip: When dealing with probability distributions, always ensure the total area under the function equals 1, as this represents the total probability.