The graph of a uniform probability distribution is given below. If necessary, round results accurate to at least 3 decimal places.

12345678910111213f(x)[Graphs generated by this script: setBorder(50,30,0,0);initPicture(0,13.9,0,1); axes(1,100,1,0,0,1,1);path([[0,0],[4,0],[4,.5],[13,.5],[13,0],[13.9,0]]);fill='red';rect([4,0],[11,.5]); text([-.5,.5],f(x));]

a. What is the probability density function (e.g., what is the **height** of the line that is the graph of the distribution)?

f(x)=   

b. Find the mean of the distribution.

μ=   

c. Find the standard deviation of the distribution.

σ=   

d. Find the probability represented by the shaded region. 
  


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.

The graph of a uniform probability distribution is given below. If necessary, round results accurate to at least 3 decimal places.
12345678910111213f(x)[Graphs generated by this script: setBorder(50,30,0,0);initPicture(0,13.9,0,1); axes(1,100,1,0,0,1,1);path([[0,0],[4,0],[4,.5],[13,.5],[13,0],[13.9,0]]);fill='red';rect([4,0],[11,.5]); text([-.5,.5],f(x));]
a. What is the probability density function (e.g., what is the height of the line that is the graph of the distribution)?
b. Find the mean of the distribution.
c. Find the standard deviation of the distribution.
d. Find the probability represented by the shaded region.

Learn how to solve a uniform probability distribution problem with a detailed step-by-step guide. This example covers calculating the probability density function, mean, standard deviation, and the probability of a shaded region. Perfect for college-level probability and statistics students.

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