To find the cumulative distribution function (CDF) based on a given probability mass function (PMF), we need to compute the cumulative probability up to each value of $x$. The CDF at a value $x$, denoted $F(x)$, is the sum of the probabilities of all values $\leq x$.

Given the PMF: $P(X = 1) = 0.1, \quad P(X = 2) = 0.4, \quad P(X = 3) = 0.3, \quad P(X = 4) = 0.2$

The CDF is calculated as follows:

1. For $x = 1$: $F(1) = P(X \leq 1) = P(X = 1) = 0.1$

2. For $x = 2$: $F(2) = P(X \leq 2) = P(X = 1) + P(X = 2) = 0.1 + 0.4 = 0.5$

3. For $x = 3$: $F(3) = P(X \leq 3) = P(X = 1) + P(X = 2) + P(X = 3) = 0.1 + 0.4 + 0.3 = 0.8$

4. For $x = 4$: $F(4) = P(X \leq 4) = P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) = 0.1 + 0.4 + 0.3 + 0.2 = 1.0$

Thus, the CDF $F(x)$ is:

\begin{cases} 0 & \text{for } x < 1 \\ 0.1 & \text{for } x = 1 \\ 0.5 & \text{for } x = 2 \\ 0.8 & \text{for } x = 3 \\ 1.0 & \text{for } x \geq 4 \end{cases}$$ Would you like more details on how cumulative distribution functions are derived, or do you have any specific questions? Here are 5 related questions to expand on this topic: 1. How does the CDF relate to the expected value of a discrete random variable? 2. What is the difference between a PMF and a PDF (Probability Density Function)? 3. Can you convert the CDF into a PMF, and if so, how? 4. How would the CDF change if one of the probabilities in the PMF were altered? 5. What is the role of the CDF in determining the median of a distribution? **Tip:** The CDF is always non-decreasing and approaches 1 as $$x$$ increases for any probability distribution.