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:

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. --- 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.