https://file.aiquickdraw.com/mathbot/file/1726376643751ovobj8lj.jpg

The given image shows a cumulative distribution function \( F(b) \) of a random variable \( X \). The function is defined piecewise as follows:

\[
F(b) =
\begin{cases}
0, & b < 0 \\
\frac{1}{2}, & 0 \leq b \leq 1 \\
1, & 1 < b < \infty
\end{cases}
\]

To find the **probability mass function** (PMF) of \( X \), we need to examine where the probability masses are concentrated. The PMF is the derivative of the cumulative distribution function (CDF) at the points of discontinuity or specific values where probabilities are assigned.

### Analysis:

- **For \( b < 0 \)**: \( F(b) = 0 \), which means there is no probability mass for \( b < 0 \).
  
- **At \( b = 0 \)**: There is a jump in the CDF from \( 0 \) to \( \frac{1}{2} \), indicating that the probability mass at \( X = 0 \) is:
  \[
  P(X = 0) = \frac{1}{2}.
  \]

- **For \( 0 < b < 1 \)**: The CDF remains constant at \( \frac{1}{2} \), meaning no probability mass is allocated for values strictly between 0 and 1.

- **At \( b = 1 \)**: The CDF jumps from \( \frac{1}{2} \) to \( 1 \), meaning there is a probability mass at \( X = 1 \):
  \[
  P(X = 1) = \frac{1}{2}.
  \]

- **For \( b > 1 \)**: The CDF remains constant at \( 1 \), so no probability mass exists for \( b > 1 \).

### Conclusion:

The probability mass function (PMF) of \( X \) is:

\[
P(X = x) =
\begin{cases}
\frac{1}{2}, & x = 0 \\
\frac{1}{2}, & x = 1 \\
0, & \text{otherwise}
\end{cases}
\]

This indicates that \( X \) is a discrete random variable with probability mass concentrated at \( x = 0 \) and \( x = 1 \), both with a probability of \( \frac{1}{2} \).

Would you like more details or have any further questions?

Here are 5 related questions to deepen your understanding:
1. How is the CDF related to the PMF for discrete random variables?
2. What is the interpretation of the CDF remaining constant between two points?
3. How do we identify points where the PMF is nonzero from a given CDF?
4. Could \( X \) be considered a Bernoulli random variable in this case? Why or why not?
5. How would the CDF look for a continuous random variable?

**Tip**: The PMF can only have values at specific points for discrete random variables, whereas the CDF gives cumulative probabilities for all values less than or equal to \( b \).

Suppose the distribution function of X is given by

F(b) =
{ 0, if b < 0
 1/2, if 0 <= b <= 1
 1, if 1 < b < infinity }

What is the probability mass function of X?

Discover how to determine the probability mass function (PMF) of a random variable X given its cumulative distribution function (CDF). This problem involves understanding key probability concepts and the relationship between CDF and PMF.

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