Pdf f(x) = 0.25 if x is between -1 and 3. It is 0 if x is otherwise. Give the cumulative distributive function. 

Given the probability density function (PDF) \( f(x) \) as:

\[
f(x) = 
\begin{cases} 
0.25 & \text{if } -1 \leq x \leq 3 \\
0 & \text{otherwise}
\end{cases}
\]

The cumulative distribution function (CDF), denoted as \( F(x) \), is obtained by integrating the PDF from the lower bound up to \( x \):

\[
F(x) = \int_{-\infty}^{x} f(t) \, dt
\]

Let's calculate \( F(x) \) piece by piece:

1. **For \( x < -1 \)**:  
   Since the PDF is 0 for \( x < -1 \), the CDF is also 0:
   \[
   F(x) = 0 \quad \text{for } x < -1
   \]

2. **For \( -1 \leq x \leq 3 \)**:  
   In this interval, \( f(x) = 0.25 \), so we integrate \( f(x) \) from \(-1\) to \(x\):
   \[
   F(x) = \int_{-1}^{x} 0.25 \, dt = 0.25(x + 1)
   \]
   Thus, the CDF in this interval is:
   \[
   F(x) = 0.25(x + 1) \quad \text{for } -1 \leq x \leq 3
   \]

3. **For \( x > 3 \)**:  
   Since the total probability must sum to 1 and the PDF is 0 for \( x > 3 \), the CDF is constant at 1:
   \[
   F(x) = 1 \quad \text{for } x > 3
   \]

Putting it all together, the cumulative distribution function \( F(x) \) is:

\[
F(x) = 
\begin{cases} 
0 & \text{if } x < -1 \\
0.25(x + 1) & \text{if } -1 \leq x \leq 3 \\
1 & \text{if } x > 3
\end{cases}
\]

Would you like more details or have any questions?

Here are five related questions:
1. How do you find the mean of a continuous random variable given its PDF?
2. What is the significance of the CDF in probability theory?
3. How would you compute the variance for the distribution given in this problem?
4. What is the interpretation of the CDF at \( x = 3 \)?
5. How would the PDF change if the interval were different, say from \( 0 \) to \( 4 \)?

**Tip:** The area under the PDF curve within a specific interval gives the probability of the random variable falling within that interval.

Learn how to calculate the cumulative distribution function (CDF) for a given probability density function (PDF) with detailed step-by-step explanations. This problem involves key concepts in probability theory, including PDF and CDF, and covers different cases for the given function. Suitable for advanced high school and undergraduate students.

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