Suppose the distribution function of X is given by

F(b) =

⎧⎪⎨

⎪⎩

0, b < 0

12

, 0≤ b < 1

1, 1 ≤ b < ∞

The given distribution function \( F(b) \) of a random variable \( X \) is piecewise and can be described as follows:

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

### Interpretation:
- For \( b < 0 \), the cumulative distribution function (CDF) \( F(b) \) is 0, meaning the probability that \( X \) is less than any value \( b < 0 \) is zero.
- For \( 0 \leq b < 1 \), \( F(b) \) is constant at \( \frac{1}{2} \), meaning the probability that \( X \) is less than \( b \) (within this range) is \( 0.5 \). This suggests that the random variable \( X \) has a point mass or discontinuity at this interval.
- For \( b \geq 1 \), \( F(b) = 1 \), meaning the probability that \( X \) is less than or equal to 1 is 1. So, after \( b = 1 \), the cumulative probability has reached its maximum.

### Implications:
- This function represents a discrete random variable because of the discontinuity and the jumps in the CDF. Specifically, \( X \) takes values at specific points rather than being continuous across an interval.
- \( X \) could possibly be a **Bernoulli** or another discrete distribution due to the step-like behavior of the function.

### Questions you may want to explore:
1. What kind of random variable is represented by this CDF (discrete, continuous, or mixed)?
2. How would the probability mass function (PMF) of \( X \) look based on this CDF?
3. What is the expectation (mean) of the random variable \( X \) given the CDF?
4. Can we find the variance of \( X \) from the given CDF?
5. How would you interpret the probability that \( X = 1 \)?

### Tip:
When analyzing a CDF, always observe the points where the function jumps to determine the discrete probabilities and focus on how the function behaves for continuous sections.

Would you like a more detailed breakdown on any of these points or another related topic?

Suppose the distribution function of X is given by F(b) = { 0, b < 0; 1/2, 0≤ b < 1; 1, 1 ≤ b < ∞ }

Explore how to analyze a cumulative distribution function (CDF) that defines a discrete random variable X with jumps at 0 and 1. This problem involves key probability concepts, including the properties of CDFs and discrete random variables. Suitable for undergraduate students studying probability theory.

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