3. Distribution Functions

3.1 Probability Mass Function (PMF)

• For: Discrete Variables. A table or function listing probabilities for specific values.

• Example (Rolling a Die):
• Let $X$ be the number shown on the die.
• PMF Table:

• Check: Sum of all probabilities $= 1$.

3.2 Probability Density Function (PDF)

• For: Continuous Variables.
• Probability is the Area Under the Curve between two points.

• Example (Bus Wait Time):
• A bus arrives every 30 minutes. Let $X$ be the wait time.
• PDF: $f(x) = 1/30$ for $0 \le x \le 30$.
• Question: What is the probability you wait less than 10 minutes?
• Solution: Area of rectangle = Width $\times$ Height.

$$ P(0 \le X \le 10) = 10 \times \frac{1}{30} = \frac{1}{3} \approx 33\% $$

3.3 Cumulative Distribution Function (CDF)

• Definition: $F(x) = P(X \le x)$. It adds up probabilities from the start up to $x$.

• Example (Die Roll):
• What is $F(3)$? (Probability of rolling a number $\\le 3$).
• Outcomes: ${1, 2, 3}$.
• $F(3) = P(1) + P(2) + P(3) = 1/6 + 1/6 + 1/6 = 3/6 = 0.5$.