01 Key Concepts
Discrete Probability Distribution
Lists every possible value of a discrete random variable alongside its probability; all probabilities must sum to 1.
Probability Mass Function (PMF)
A function P(X=x) that gives the probability for each specific value x of a discrete random variable.
Probability Density Function (PDF)
For continuous variables, describes relative likelihood; probabilities come from the area under the curve over an interval, not from a single point.
Cumulative Distribution Function (CDF)
F(x) = P(X <= x), giving the probability that X takes a value less than or equal to x.
Requirements for a Valid Distribution
Every probability must be between 0 and 1, and the sum (discrete) or total area (continuous) must equal exactly 1.
02 Key Formulas
- Sum of P(x) over all x = 1 (discrete case)
- F(x) = P(X <= x)
03 Solved Examples
Example 1 Verify that P(X=1)=0.2, P(X=2)=0.5, P(X=3)=0.3 is a valid probability distribution.
- Check each value is between 0 and 1: yes for all three.
- Sum them: 0.2 + 0.5 + 0.3.
Answer: Sum = 1.0, so it is a valid distribution
Example 2 For the distribution above, find F(2), the cumulative probability up to X=2.
- F(2) = P(X<=2) = P(X=1) + P(X=2) = 0.2 + 0.5.
Answer: F(2) = 0.7
Example 3 Is P(X=1)=0.5, P(X=2)=0.6 a valid probability distribution?
- Sum: 0.5 + 0.6 = 1.1, which is greater than 1.
Answer: No, the probabilities exceed 1, so it is invalid
04 Practice Questions
1Is P(X=1)=0.3, P(X=2)=0.7 a valid distribution?
Yes, sums to 1
2Is P(X=1)=-0.2, P(X=2)=1.2 valid?
No, probabilities cannot be negative
3For P(X=1)=0.4, P(X=2)=0.6, find F(1).
0.4
4What must the sum of all probabilities equal in a discrete distribution?
1
5What does a CDF, F(x), represent?
The probability that X is less than or equal to x
📄 Probability Distributions — Downloadable Worksheet
10 questions with a full answer key. Grab the PDF to print, or try the interactive version in your browser.