if pmf is given in the style of cdf then how to find k

To find the value of \( k \) when a probability mass function (PMF) is given in the style of a cumulative distribution function (CDF), you should follow these steps:

### 1. **Understand the CDF and PMF Relationship:**
   - The CDF, \( F(x) \), gives the probability that a random variable \( X \) takes a value less than or equal to \( x \): 
     \[
     F(x) = P(X \leq x)
     \]
   - The PMF, \( p(x) \), gives the probability that \( X \) takes the value \( x \):
     \[
     p(x) = P(X = x)
     \]
   - The relationship between the PMF and the CDF is:
     \[
     p(x) = F(x) - F(x-1)
     \]
     This is the difference between the CDF values at \( x \) and \( x-1 \).

### 2. **Set Up the Equation to Find \( k \):**
   - If the PMF is given in the style of the CDF, first express the PMF \( p(x) \) in terms of \( k \). Often, the CDF or the PMF itself will involve \( k \) as an unknown constant.
   - Sum all the PMF values over the support of the random variable. Since the total probability must equal 1, set up the equation:
     \[
     \sum_{x} p(x) = 1
     \]

### 3. **Solve for \( k \):**
   - Substitute the expression for \( p(x) \) into the summation.
   - Solve the equation to find the value of \( k \).

### **Example Problem:**

Suppose the CDF is given by:
\[
F(x) = \begin{cases} 
0 & x < 1 \\
\frac{k}{4} & 1 \leq x < 2 \\
\frac{3k}{4} & 2 \leq x < 3 \\
1 & x \geq 3 
\end{cases}
\]
To find \( k \), follow these steps:

1. **Find the PMF using the CDF:**
   - For \( x = 1 \): \( p(1) = F(1) - F(0) = \frac{k}{4} - 0 = \frac{k}{4} \)
   - For \( x = 2 \): \( p(2) = F(2) - F(1) = \frac{3k}{4} - \frac{k}{4} = \frac{2k}{4} = \frac{k}{2} \)
   - For \( x = 3 \): \( p(3) = F(3) - F(2) = 1 - \frac{3k}{4} = \frac{4 - 3k}{4} \)

2. **Set up the normalization condition:**
   \[
   p(1) + p(2) + p(3) = \frac{k}{4} + \frac{k}{2} + \frac{4 - 3k}{4} = 1
   \]

3. **Solve for \( k \):**
   \[
   \frac{k}{4} + \frac{2k}{4} + \frac{4 - 3k}{4} = 1
   \]
   Simplifying:
   \[
   \frac{k + 2k + 4 - 3k}{4} = 1 \implies \frac{4}{4} = 1
   \]
   Since the equation is true, it implies \( k = 1 \).

So, \( k = 1 \) in this case.

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Do you want more details, or have any questions about this process? 

Here are 5 related questions you might find interesting:
1. How do you find the CDF if you are given the PMF directly?
2. What is the relationship between the PMF and the expectation of a discrete random variable?
3. How do you find the median of a discrete random variable using the CDF?
4. How can the CDF be used to find the probability of a range of values?
5. How does the CDF change if the distribution is continuous instead of discrete?

**Tip:** Always verify that the sum of the PMF equals 1, as it ensures the validity of the probability distribution.

Learn how to find the value of k when a probability mass function (PMF) is given in the style of a cumulative distribution function (CDF). This detailed guide covers the relationship between PMF and CDF, setting up equations, and solving for the unknown constant k. Suitable for advanced undergraduate students and beyond.

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