Solved examples: Deriving PMF from CDF for discrete random variables.

📚 Quick Study Guide

• 📊 The Cumulative Distribution Function (CDF), $F_X(x)$, gives the probability that the random variable $X$ takes on a value less than or equal to $x$. Mathematically, $F_X(x) = P(X \le x)$.
• 🔢 The Probability Mass Function (PMF), $P_X(x)$, gives the probability that the random variable $X$ takes on a specific value $x$. Mathematically, $P_X(x) = P(X = x)$.
• 🧮 For a discrete random variable, you can derive the PMF from the CDF using the following relationship: $P_X(x) = F_X(x) - F_X(x-1)$. This means the probability of $X$ being exactly $x$ is the difference between the cumulative probability up to $x$ and the cumulative probability up to $x-1$.
• 💡 Remember that $F_X(-\infty) = 0$ and $F_X(\infty) = 1$. Also, the PMF values must sum to 1: $\sum_{x} P_X(x) = 1$.

Practice Quiz

1. Question 1: Given the CDF of a discrete random variable $X$ is $F_X(2) = 0.3$ and $F_X(1) = 0.1$, what is $P_X(2)$?
   1. A) 0.1
   2. B) 0.2
   3. C) 0.3
   4. D) 0.4
2. Question 2: If $F_X(5) = 0.8$ and $F_X(4) = 0.6$, find $P_X(5)$.
   1. A) 0.2
   2. B) 0.4
   3. C) 0.6
   4. D) 0.8
3. Question 3: Suppose $F_X(0) = 0.2$, $F_X(1) = 0.5$, and $F_X(2) = 1.0$. What is $P_X(1)$?
   1. A) 0.2
   2. B) 0.3
   3. C) 0.5
   4. D) 1.0
4. Question 4: A discrete random variable has CDF $F_X(3) = 0.4$ and $F_X(2) = 0.1$. Determine $P_X(3)$.
   1. A) 0.1
   2. B) 0.2
   3. C) 0.3
   4. D) 0.4
5. Question 5: Given $F_X(-1) = 0.0$, $F_X(0) = 0.4$, and $F_X(1) = 0.9$, calculate $P_X(0)$.
   1. A) 0.0
   2. B) 0.4
   3. C) 0.5
   4. D) 0.9
6. Question 6: If $F_X(7) = 0.9$ and $F_X(6) = 0.7$, what is the value of $P_X(7)$?
   1. A) 0.2
   2. B) 0.7
   3. C) 0.9
   4. D) 1.6
7. Question 7: The CDF of a discrete random variable is $F_X(4) = 0.5$ and $F_X(3) = 0.2$. Find the PMF at $x = 4$, i.e., $P_X(4)$.
   1. A) 0.2
   2. B) 0.3
   3. C) 0.5
   4. D) 0.7