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Let's solve the given problems step by step.

### Problem Statement:
We are given three probability density functions (PDFs) \( f(x) \) and asked to find the cumulative distribution function (CDF) \( F(x) \) associated with each, and sketch their graphs.

The CDF \( F(x) \) is defined as:
\[
F(x) = \int_{-\infty}^{x} f(t) \, dt
\]

We will find the CDF for each part.

---

### (a) \( f(x) = 3(1 - x)^2 \), for \( 0 < x < 1 \), zero elsewhere.
1. **CDF:**
   \[
   F(x) = \int_{0}^{x} 3(1 - t)^2 \, dt
   \]
   Let's compute the integral.

   \[
   \int 3(1 - t)^2 \, dt = 3 \int (1 - 2t + t^2) \, dt = 3 \left( t - t^2 + \frac{t^3}{3} \right)
   \]
   Now, evaluate from 0 to \( x \):
   \[
   F(x) = 3 \left( x - x^2 + \frac{x^3}{3} \right) - 3(0) = 3x - 3x^2 + x^3
   \]
   Therefore, the CDF is:
   \[
   F(x) = 3x - 3x^2 + x^3 \quad \text{for} \, 0 < x < 1
   \]
   For \( x \leq 0 \), \( F(x) = 0 \) (since \( f(x) = 0 \)).
   For \( x \geq 1 \), \( F(x) = 1 \) (since it's the total probability).
   
---

### (b) \( f(x) = \frac{1}{x^2} \), for \( 1 < x < \infty \), zero elsewhere.
1. **CDF:**
   \[
   F(x) = \int_{1}^{x} \frac{1}{t^2} \, dt
   \]
   Let's compute the integral.

   \[
   \int \frac{1}{t^2} \, dt = -\frac{1}{t}
   \]
   Now, evaluate from 1 to \( x \):
   \[
   F(x) = -\frac{1}{x} + 1
   \]
   Therefore, the CDF is:
   \[
   F(x) = 1 - \frac{1}{x} \quad \text{for} \, x > 1
   \]
   For \( x \leq 1 \), \( F(x) = 0 \) (since \( f(x) = 0 \)).

---

### (c) \( f(x) = \frac{1}{3} \), for \( 0 < x < 1 \) or \( 2 < x < 4 \), zero elsewhere.
1. **CDF:**

   We need to handle this in two intervals:
   - For \( 0 < x < 1 \):
     \[
     F(x) = \int_0^x \frac{1}{3} \, dt = \frac{x}{3}
     \]
   - For \( 1 \leq x < 2 \), \( f(x) = 0 \), so \( F(x) = \frac{1}{3} \) (since no probability is added).
   - For \( 2 < x < 4 \):
     \[
     F(x) = \frac{1}{3} + \int_2^x \frac{1}{3} \, dt = \frac{1}{3} + \frac{x - 2}{3}
     \]
     Simplifying this:
     \[
     F(x) = \frac{x + 1}{3}
     \]

Therefore, the CDF for this case is:
   \[
   F(x) =
   \begin{cases}
   0 & \text{if} \, x \leq 0 \\
   \frac{x}{3} & \text{if} \, 0 < x < 1 \\
   \frac{1}{3} & \text{if} \, 1 \leq x \leq 2 \\
   \frac{x + 1}{3} & \text{if} \, 2 < x < 4 \\
   1 & \text{if} \, x \geq 4
   \end{cases}
   \]

---

### Summary:
- (a) \( F(x) = 3x - 3x^2 + x^3 \), for \( 0 < x < 1 \).
- (b) \( F(x) = 1 - \frac{1}{x} \), for \( x

Find the CDF F(x) associated with each of the following probability density functions. Sketch the graphs of f(x) and F(x). (a) f(x) = 3(1 - x)^2 for 0 < x < 1, zero elsewhere. (b) f(x) = 1/x^2 for 1 < x < ∞, zero elsewhere. (c) f(x) = 1/3 for 0 < x < 1 or 2 < x < 4, zero elsewhere.

Explore the step-by-step solution for finding the CDF associated with probability density functions. This solution covers integration techniques for f(x) = 3(1 - x)^2, f(x) = 1/x^2, and a piecewise function f(x) = 1/3. Suitable for advanced high school or undergraduate students studying probability theory.

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