A spinner from a board game randomly indicates a real number between 0 and 20. The spinner is fair in the sense that it indicates a number in a given interval with the same probability as it indicates a number in any other interval of the same length.
(a) Explain why the function
f(x) = 

0.05  	if 	0 ≤ x ≤ 20
0	if	x < 0 or x > 20
is a probability density function for the spinner's values.

To explain why the function 

\[ f(x) = \begin{cases} 
0.05 & \text{if } 0 \le x \le 20 \\ 
0 & \text{if } x < 0 \text{ or } x > 20 
\end{cases} \]

is a probability density function (pdf) for the spinner's values, we need to verify that it satisfies the two main properties of a probability density function:

1. The function \( f(x) \) must be non-negative for all \( x \).
2. The total area under the curve of \( f(x) \) over its entire range must be equal to 1.

### Property 1: Non-negativity

For \( f(x) \) to be a valid pdf, it must be non-negative for all \( x \). The given function \( f(x) \) is defined as:

\[ f(x) = \begin{cases} 
0.05 & \text{if } 0 \le x \le 20 \\ 
0 & \text{if } x < 0 \text{ or } x > 20 
\end{cases} \]

In this definition, \( f(x) \) is either 0.05 (which is positive) within the interval [0, 20], or 0 outside this interval. Therefore, \( f(x) \) is non-negative for all \( x \), satisfying the first property of a pdf.

### Property 2: Total Area Under the Curve

The total area under the curve of \( f(x) \) must equal 1. This is calculated by integrating \( f(x) \) over its entire range:

\[ \int_{-\infty}^{\infty} f(x) \, dx \]

Given that \( f(x) \) is only non-zero between 0 and 20, we can limit the integral to this interval:

\[ \int_{0}^{20} 0.05 \, dx \]

Evaluate the integral:

\[ \int_{0}^{20} 0.05 \, dx = 0.05 \int_{0}^{20} dx = 0.05 [x]_{0}^{20} = 0.05 (20 - 0) = 0.05 \times 20 = 1 \]

The total area under the curve is 1, satisfying the second property of a pdf.

### Conclusion

Since the function \( f(x) \) is non-negative for all \( x \) and the total area under the curve is 1, the function:

\[ f(x) = \begin{cases} 
0.05 & \text{if } 0 \le x \le 20 \\ 
0 & \text{if } x < 0 \text{ or } x > 20 
\end{cases} \]

is a valid probability density function for the spinner's values.

Do you want details or have any questions?

---

1. What is a probability density function (pdf)?
2. How is the area under the curve of a pdf related to probabilities?
3. Why must the total area under a pdf equal 1?
4. How do you interpret a uniform distribution?
5. What is the significance of the function being non-negative?
6. Can you provide another example of a pdf?
7. How would the pdf change if the spinner indicated values between different limits, e.g., 0 and 30?
8. What are some practical applications of pdfs in real life?

**Tip:** Always verify that the total area under a pdf equals 1, as this ensures the function properly represents a probability distribution.

Explore how the function f(x) = 0.05 for 0 ≤ x ≤ 20 serves as a probability density function (pdf) for a spinner game. Learn about the properties of pdfs, including non-negativity and the total area under the curve equating to 1.

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