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.