The problem involves a uniform distribution, which is a type of probability distribution where all outcomes are equally likely within a certain range. Let's break down and solve both parts of the question.

Part (a) What is the height of the function, $f(x)$?

• The uniform distribution has constant height $f(x)$ over a given interval.
• For a uniform distribution over an interval $[a, b]$, the total area under the curve must equal 1 (since the total probability is always 1).

Thus, the height $f(x)$ is calculated by:

$f(x) = \frac{1}{b - a}$

From the diagram, the uniform distribution spans the interval $[0, 12]$. So, $a = 0$ and $b = 12$.

$f(x) = \frac{1}{12 - 0} = \frac{1}{12}$

So, the height of the function is $\frac{1}{12}$.

Part (b) What is the probability of the shaded region?

• The shaded region corresponds to the interval from $x = 9$ to $x = 12$.
• The probability of the shaded region is the area under the curve from 9 to 12.

Since the height $f(x) = \frac{1}{12}$, the probability for any interval $[x_1, x_2]$ within the uniform distribution is given by:

$\text{Probability} = f(x) \times (\text{Length of the interval})$

In this case, the interval is from $9$ to $12$, so the length is:

$12 - 9 = 3$

Therefore, the probability of the shaded region is:

$\text{Probability} = \frac{1}{12} \times 3 = \frac{3}{12} = \frac{1}{4}$

Final Answers:

a) The height of the function $f(x) = \frac{1}{12}$.

b) The probability of the shaded region is $\frac{1}{4}$.

Do you want more details or have any further questions?

Here are 5 related questions to expand understanding:

1. How would the height of $f(x)$ change if the interval was $[0, 15]$?
2. What is the probability of the interval between $x = 3$ and $x = 8$?
3. How can you calculate the expected value (mean) of a uniform distribution?
4. What would the probability be if the shaded region was $x = 7$ to $x = 12$?
5. How would you calculate the variance of this uniform distribution?

Tip: The total area under the probability density function (PDF) of any distribution always equals 1, as it represents the total probability.