# Section 3: Discrete Random Variable, Probability Mass Function, and Cumulative Distribution Function

### Instructions

• This section covers the concepts listed below.
• For each concept, there is a conceptual video explaining it followed by videos working through examples.
• When you have finished the material below, you can go to the next section or return to the main Mathematical Probability page
Discrete Random Variable

### Examples

Directions: There are no examples for this video.

### Self-Assessment Questions

Directions: The following questions are an assessment of your understanding of the material above. If you are not sure of the answers, you may need to rewatch the videos.
1. What is a random variable? What makes it random as opposed to deterministic?
2. What is a discrete random variable? Give an example.
3. Is the following continuous or discrete? The weight of a newborn baby? Measured weight of a newborn baby in grams?
4. Is a random variable either discrete or continuous?
Probability Distribution and Probability Mass Function

### Examples

Directions: The following examples cover the material from the video above.

### Self-Assessment Questions

Directions: The following questions are an assessment of your understanding of the material above. If you are not sure of the answers, you may need to rewatch the videos.
1. What is the meaning of a probability distribution in general? How do we realize it practically for discrete random variables?
2. Let $$X$$ be a discrete random variable with probability mass function $$p_X$$. What does it mean to have $$p_X(3)=0$$?
Functions of Random Variables

### Examples

Directions: The following examples cover the material from the video above.

### Self-Assessment Questions

Directions: The following questions are an assessment of your understanding of the material above. If you are not sure of the answers, you may need to rewatch the videos.
1. Give a concrete example of a random experiment, a random variable $$X$$ on it, and a function of $$X$$ as another random variable.
2. Is every function of a discrete random variable discrete? If not, provide a counterexample.
3. Is every function of a continuous random variable continuous? If not, provide a counterexample.
Cumulative Distribution Function

### Examples

Directions: The following examples cover the material from the video above.

### Self-Assessment Questions

Directions: The following questions are an assessment of your understanding of the material above. If you are not sure of the answers, you may need to rewatch the videos.
1. Define the CDF of a random variable.
2. What is the unifying aspect of the CDF’s?
3. List all 4 properties of the CDF's.
4. If the CDF of a random variable is given as $$F_X$$, can you use that to find $$P(1<X\leq 2)$$, $$P(X=2)$$, and $$P(X\geq 2)$$?