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How would I solve this problem? do not solve the problem, only show me the steps

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 \( p(1 \leq X \leq 3) \).

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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 \( \sum p(X=k) = 1 \).

A random variable X has PMF p(X = k) = a / [k(k+1)] for k = 1, 2, ..., 4, where a is a constant. Then, what is the value of p(1 ≤ X ≤ 3)? Round to 3 decimal places.

Learn how to compute p(1 ≤ X ≤ 3) for a random variable X with PMF p(X = k) = a / [k(k+1)] for k = 1, 2, ..., 4. This solution covers discrete random variables and how to normalize probabilities for a PMF.

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