Let's break down the components of the problem based on the given graph and information. It appears to be a uniform probability distribution defined over a certain interval, and we're tasked with calculating several values from it.

a. Probability Density Function (PDF)

The height of the line for a uniform distribution is the value of the probability density function $f(x)$. For a uniform distribution over an interval $[a, b]$, the PDF is defined as:

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

From the graph:

• The distribution starts at $x = 4$ and ends at $x = 13$.

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

$f(x) = \frac{1}{13 - 4} = \frac{1}{9} \approx 0.111$

b. Mean of the Distribution (μ)

For a uniform distribution over an interval $[a, b]$, the mean $\mu$ is the midpoint of the interval:

$\mu = \frac{a + b}{2}$

Using $a = 4$ and $b = 13$:

$\mu = \frac{4 + 13}{2} = \frac{17}{2} = 8.5$

c. Standard Deviation of the Distribution (σ)

The standard deviation $\sigma$ for a uniform distribution over $[a, b]$ is given by the formula:

$\sigma = \frac{b - a}{\sqrt{12}}$

Substituting $a = 4$ and $b = 13$:

$\sigma = \frac{13 - 4}{\sqrt{12}} = \frac{9}{\sqrt{12}} \approx \frac{9}{3.464} \approx 2.597$

d. Probability of the Shaded Region

In a uniform distribution, the probability of a region between two points $c$ and $d$, where $c \leq d$, is given by:

$P(c \leq x \leq d) = \frac{d - c}{b - a}$

To find the exact probability, we'd need to know the boundaries of the shaded region on the graph. If you have that information, I can calculate this probability more specifically.

Would you like further clarification or details?

Related Questions:

1. How is the height of the PDF determined for a uniform distribution?
2. What does the standard deviation represent in a uniform distribution?
3. Why is the mean simply the midpoint of the interval in a uniform distribution?
4. How can we visually interpret the probability of a region in a uniform distribution graph?
5. What would happen to the standard deviation if the interval $[a, b]$ were wider?

Tip:

For a uniform distribution, the area under the curve between two points gives the probability for that interval. The total area under the curve over the entire distribution is always 1.