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The graph provided represents the cumulative distribution function (CDF) of a discrete random variable \( X \). The CDF gives the probability that \( X \) takes a value less than or equal to a certain value, i.e., \( F(x) = P(X \leq x) \).

Let's go through each question using the information from the CDF.

### a) \( P(X = -4) \)

To find \( P(X = -4) \), we need to determine the difference in the CDF at \( x = -4 \). 
- From the graph, at \( x = -4 \), \( F(-4) = 0.1 \).
- Immediately before \( x = -4 \), \( F(x) = 0 \).
  
Thus, \( P(X = -4) = F(-4) - F(\text{just before } -4) = 0.1 - 0 = 0.1 \).

### b) \( P(X = -2) \)

For \( P(X = -2) \), again, we calculate the difference in the CDF at \( x = -2 \).
- From the graph, at \( x = -2 \), \( F(-2) = 0.3 \).
- Just before \( x = -2 \), \( F(x) = 0.1 \).

So, \( P(X = -2) = 0.3 - 0.1 = 0.2 \).

### c) \( P(X = 0) \)

Similarly, we look at the change in the CDF at \( x = 0 \).
- From the graph, at \( x = 0 \), \( F(0) = 0.5 \).
- Just before \( x = 0 \), \( F(x) = 0.3 \).

Thus, \( P(X = 0) = 0.5 - 0.3 = 0.2 \).

### d) \( P(X \leq 0) \)

\( P(X \leq 0) \) is simply the value of the CDF at \( x = 0 \).
- From the graph, \( F(0) = 0.5 \).

Thus, \( P(X \leq 0) = 0.5 \).

### e) \( P(X > 3) \)

To find \( P(X > 3) \), we need to subtract \( F(3) \) from 1.
- From the graph, \( F(3) = 1 \), which means the CDF has reached its maximum value at \( x = 3 \).

So, \( P(X > 3) = 1 - F(3) = 1 - 1 = 0 \).

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Would you like more details or have any questions about these answers?

### Related Questions:
1. How is a CDF different from a probability mass function (PMF)?
2. What does the CDF represent for a continuous random variable?
3. How would \( P(X = 3) \) be calculated from the CDF graph?
4. Why is \( P(X > 3) \) equal to zero in this case?
5. How would the probabilities change if the graph showed a continuous random variable?

### Tip:
The CDF is always a non-decreasing function, meaning that as \( x \) increases, the probability that \( X \leq x \) also increases or stays the same.

Let X be a discrete random variable. Suppose the graph of the CDF of X is given above. Use the graph to answer the following questions:

a) P(X = -4).
b) P(X = -2).
c) P(X = 0).
d) P(X ≤ 0).
e) P(X > 3).

Learn how to calculate specific probabilities from the CDF of a discrete random variable using the CDF graph. This detailed solution includes step-by-step calculations for P(X = -4), P(X = -2), P(X = 0), P(X ≤ 0), and P(X > 3). Suitable for students in high school or introductory college-level probability courses.

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