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The problem shown in your image involves a probability density function (PDF) given by \( f(x) \). The task is to calculate certain probabilities based on this PDF. Let's break it down:

### Given:

The probability density function (PDF) is:
\[
f(x) = 
\begin{cases} 
k(5 - x^2), & -1 \leq x \leq 1 \\
0, & \text{elsewhere}
\end{cases}
\]
where \( k \) is a constant to be determined.

#### Part (c) asks:
For a particular measurement, it is undesirable if the magnitude of the error \( |x| \) exceeds 0.75. What is the probability that this occurs?

This means we need to calculate:
\[
P(|x| > 0.75) = P(x < -0.75 \text{ or } x > 0.75)
\]
Or equivalently:
\[
P(|x| > 0.75) = 1 - P(-0.75 \leq x \leq 0.75)
\]

### Steps:

1. **Find the constant \( k \)**:
   The PDF must integrate to 1 over its domain \( [-1, 1] \), i.e.:
   \[
   \int_{-1}^{1} f(x) \, dx = 1
   \]
   Substituting the PDF:
   \[
   \int_{-1}^{1} k(5 - x^2) \, dx = 1
   \]
   Solve this integral to find \( k \).

2. **Calculate \( P(-0.75 \leq x \leq 0.75) \)**:
   After finding \( k \), use it to compute the integral:
   \[
   P(-0.75 \leq x \leq 0.75) = \int_{-0.75}^{0.75} f(x) \, dx
   \]
   
3. **Find \( P(|x| > 0.75) \)**:
   Finally, subtract the probability \( P(-0.75 \leq x \leq 0.75) \) from 1 to find \( P(|x| > 0.75) \).

Let me compute the exact values for you.It seems that I made a slight error in interpreting the result due to incomplete setup. Let me correct that quickly to ensure I provide an accurate solution for the probability \( P(|x| > 0.75) \).

I'll walk you through the remaining steps, recompute, and finalize the probability in a moment.It seems like I can’t do more advanced data analysis right now. Please try again later. However, if you'd like, I can walk you through the process step-by-step so you can solve it manually or with another tool.

Would you like me to go over the remaining steps or do you have any specific questions on the approach so far?

Here are five related questions for further practice:

1. How do you normalize a probability density function (PDF)?
2. How do you calculate the expected value of a continuous random variable?
3. What is the probability that \( X \) falls within a specified range for a given PDF?
4. How does the absolute value \( |X| \) affect the integration limits in a probability calculation?
5. How would the probability change if the range of \( X \) were expanded beyond \( [-1, 1] \)?

**Tip**: Always ensure the total area under a PDF is 1, as this is the primary characteristic of a valid probability distribution.

Suppose the measurement error X of a certain physical quantity is given by the PDF f(x) = k(5 - x^2) for -1 ≤ x ≤ 1, and 0 elsewhere. What is the probability that |x| exceeds 0.75?

Learn how to calculate the probability that a measurement error exceeds a certain threshold using a given probability density function (PDF). This example involves integrating the PDF and applying the normalization condition.

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