https://file.aiquickdraw.com/mathbot/file/1728148006675nunemog9.jpg

To solve the given problem based on the image:

We are given a probability distribution with the random variable \( X \) taking values \( -2, -1, 0, 1, 2 \) with associated probabilities \( f(x) \) as follows:

\[
X = -2, -1, 0, 1, 2
\]
\[
f(X) = 0.125, 0.25, 0.25, 0.25, 0.125
\]

Now, let's address each probability question:

1. **\( P(X \leq 2) \):**
   Since 2 is the maximum value of \( X \), \( P(X \leq 2) \) includes all probabilities, so:

   \[
   P(X \leq 2) = 0.125 + 0.25 + 0.25 + 0.25 + 0.125 = 1
   \]
   Thus, \( P(X \leq 2) = 1 \).

2. **\( P(X > -2) \):**
   This excludes \( X = -2 \), so:

   \[
   P(X > -2) = P(X = -1) + P(X = 0) + P(X = 1) + P(X = 2)
   \]
   \[
   P(X > -2) = 0.25 + 0.25 + 0.25 + 0.125 = 0.875
   \]

3. **\( P(-1 \leq X \leq 1) \):**
   This includes \( X = -1, 0, 1 \), so:

   \[
   P(-1 \leq X \leq 1) = P(X = -1) + P(X = 0) + P(X = 1)
   \]
   \[
   P(-1 \leq X \leq 1) = 0.25 + 0.25 + 0.25 = 0.75
   \]

4. **\( P(X < -1) \):**
   This only includes \( X = -2 \), so:

   \[
   P(X < -1) = P(X = -2) = 0.125
   \]

### Summary:
- \( P(X \leq 2) = 1 \)
- \( P(X > -2) = 0.875 \)
- \( P(-1 \leq X \leq 1) = 0.75 \)
- \( P(X < -1) = 0.125 \)

Would you like more details or explanations for any of these steps?

Here are some related questions you might find useful:
1. What is the expected value \( E(X) \) of this probability distribution?
2. How do we calculate the variance of \( X \)?
3. What is the cumulative distribution function (CDF) of \( X \)?
4. How do you find \( P(0 \leq X \leq 2) \)?
5. How would you graph this probability mass function?

**Tip**: The sum of all probabilities in any discrete probability distribution should always equal 1. This is a good way to check the consistency of a distribution.

Determine the following probabilities: P(X <= 2), P(X > -2), P(-1 <= X <= 1), P(X < -1).

Learn how to determine probabilities from a discrete probability distribution where X takes values -2, -1, 0, 1, and 2. This covers calculating cumulative probabilities such as P(X <= 2), P(X > -2), and other key statistics concepts. Suitable for high school or college-level statistics students.

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