To solve this problem, where you are asked to find $p(1 \leq X \leq 3)$, follow these steps:

Step 1: Identify the Probability Mass Function (PMF)

You are given the PMF for the random variable $X$: $p(X = k) = \frac{a}{k(k+1)}, \quad k = 1, 2, 3, 4$ where $a$ is a constant.

Step 2: Find the Range of Interest

You are asked to compute the probability that $X$ is between 1 and 3, inclusive: $p(1 \leq X \leq 3) = p(X = 1) + p(X = 2) + p(X = 3)$ This means you will sum the probabilities of $X$ being equal to 1, 2, and 3 using the given PMF formula.

Step 3: Compute Each Individual Probability

For each $k = 1, 2, 3$, calculate the probability: $p(X = k) = \frac{a}{k(k+1)}$ Substitute $k = 1, 2, 3$ into the PMF formula to find:

• $p(X = 1)$
• $p(X = 2)$
• $p(X = 3)$

Step 4: Normalize the PMF

Since the total probability must sum to 1, use the fact that: $p(X = 1) + p(X = 2) + p(X = 3) + p(X = 4) = 1$ This equation allows you to solve for the constant $a$.

Step 5: Calculate the Final Probability

Once you have the value of $a$, use it to calculate $p(1 \leq X \leq 3)$.

Let me know if you would like further clarifications or want to proceed with specific parts of the solution!

Here are 5 follow-up questions:

1. How is the constant $a$ determined in a probability mass function?
2. Why does the sum of all probabilities for a discrete random variable equal 1?
3. What are the conditions for a function to be a valid probability mass function (PMF)?
4. Can you extend this method to find probabilities for different ranges of $X$?
5. How does the PMF change if the range of $k$ increases?

Tip: When solving for a constant in probability problems, always check the total probability constraint $\sum p(X=k) = 1$.