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The problem asks to study the concepts of random variables, distribution functions, and density functions. Let's break down the problem step by step.

### Given:

The cumulative distribution function (CDF) \( F(t) \) of a continuous random variable \( X \) is defined as:

\[
F(t) =
\begin{cases} 
0, & t \leq 0 \\
ct, & 0 \leq t \leq 1 \\
1, & t > 1 
\end{cases}
\]

where \( c \) is a constant.

### Part (a): 
Determine the constant \( c \) and describe the probability density function \( f(t) \).

1. **To find \( c \)**:  
   Since \( F(t) \) is a cumulative distribution function (CDF), it must satisfy the condition that \( F(t) \to 1 \) as \( t \to \infty \). From the given definition of \( F(t) \), we know that \( F(t) = 1 \) for \( t > 1 \). Therefore, we need \( F(1) = 1 \), which gives us:
   
   \[
   F(1) = c \cdot 1 = 1
   \]
   Solving for \( c \):
   \[
   c = 1
   \]

2. **To describe the probability density function (PDF) \( f(t) \)**:  
   The probability density function is the derivative of the cumulative distribution function:
   \[
   f(t) = \frac{d}{dt}F(t)
   \]

   Now, calculate the derivative for each interval:

   - For \( t \leq 0 \), \( F(t) = 0 \), so \( f(t) = 0 \).
   - For \( 0 \leq t \leq 1 \), \( F(t) = t \), so \( f(t) = \frac{d}{dt}(t) = 1 \).
   - For \( t > 1 \), \( F(t) = 1 \), so \( f(t) = 0 \).

   Therefore, the probability density function \( f(t) \) is:

   \[
   f(t) =
   \begin{cases}
   0, & t \leq 0 \\
   1, & 0 < t < 1 \\
   0, & t \geq 1
   \end{cases}
   \]

### Part (b):
Calculate the probabilities:
1. \( P(X = \frac{1}{3}) \)
2. \( P(X > \frac{1}{3}) \)
3. \( P(|X| < \frac{1}{3}) \)

1. **For \( P(X = \frac{1}{3}) \)**:  
   Since \( X \) is a continuous random variable, the probability of \( X \) taking any single specific value is zero:
   \[
   P(X = \frac{1}{3}) = 0
   \]

2. **For \( P(X > \frac{1}{3}) \)**:  
   This probability can be found using the CDF \( F(t) \):
   \[
   P(X > \frac{1}{3}) = 1 - F\left(\frac{1}{3}\right)
   \]

   From the CDF definition, for \( t = \frac{1}{3} \), we have:
   \[
   F\left(\frac{1}{3}\right) = \frac{1}{3}
   \]
   Therefore:
   \[
   P(X > \frac{1}{3}) = 1 - \frac{1}{3} = \frac{2}{3}
   \]

3. **For \( P(|X| < \frac{1}{3}) \)**:  
   This is equivalent to \( P(-\frac{1}{3} < X < \frac{1}{3}) \). Since the random variable is non-negative (based on the CDF definition), we only consider the interval \( 0 < X < \frac{1}{3} \):
   \[
   P(0 < X < \frac{1}{3}) = F\left(\frac{1}{3}\right) - F(0)
   \]
   From the CDF definition:
   \[
   F(0) = 0, \quad F\left(\frac{1}{3}\right) = \frac{1}{3}
   \]
   Therefore:
   \[
   P(|X| < \frac{1}{3}) = \frac{1}{3} - 0 = \frac{1}{3}
   \]

---

### Summary of Results:

1. The constant \( c = 1 \).
2. The probability density function \( f(t) \) is:
   \[
   f(t) =
   \begin{cases}
   0, & t \leq 0 \\
   1, & 0 < t < 1 \\
   0, & t \geq 1
   \end{cases}
   \]
3. The probabilities are:
   - \( P(X = \frac{1}{3}) = 0 \)
   - \( P(X > \frac{1}{3}) = \frac{2}{3} \)
   - \( P(|X| < \frac{1}{3}) = \frac{1}{3} \)

---

Would you like more details or have any questions?

Here are some related questions:
1. How is the cumulative distribution function different from the probability density function?
2. What are the properties of a continuous random variable's CDF and PDF?
3. How do you interpret probabilities for a continuous random variable?
4. How do we calculate probabilities from a given PDF?
5. What happens to the PDF when the CDF is not continuous?

**Tip**: The area under the PDF curve between two points gives the probability that the random variable lies between those points.

Estudiar los conceptos de variable aleatoria, función de distribución y funciones de densidad.

Problema 1:

Una variable aleatoria X tiene función de distribución continua F, siendo 
F(t) =
0, si t ≤ 0 
ct, si 0 ≤ t ≤ 1
1, si t > 1
a. Determinar la constantey describir la función densidad f
b. Calcular las probabilidades P(X = 1/3), P(X > 1/3), P(|X| < 1/3)

Learn how to determine the constant for a given cumulative distribution function and describe the probability density function. Calculate probabilities for specific events with detailed solutions. Suitable for college or university students studying probability theory.

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