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Probability · Quick Reference
Probability Distributions Cheat Sheet
In one line: a probability distribution describes how probabilities are spread across all the possible values of a random variable — the complete 'shape' of randomness for that variable.
Key Ideas
1Discrete Probability Distribution. Lists every possible value of a discrete random variable alongside its probability; all probabilities must sum to 1.
2Probability Mass Function (PMF). A function P(X=x) that gives the probability for each specific value x of a discrete random variable.
3Probability 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.
4Cumulative Distribution Function (CDF). F(x) = P(X <= x), giving the probability that X takes a value less than or equal to x.
5Requirements for a Valid Distribution. Every probability must be between 0 and 1, and the sum (discrete) or total area (continuous) must equal exactly 1.
Worked Examples
Verify that P(X=1)=0.2, P(X=2)=0.5, P(X=3)=0.3 is a valid probability distribution.
Sum = 1.0, so it is a valid distribution
For the distribution above, find F(2), the cumulative probability up to X=2.
F(2) = 0.7
Is P(X=1)=0.5, P(X=2)=0.6 a valid probability distribution?
No, the probabilities exceed 1, so it is invalid
Formulas
Sum of P(x) over all x = 1 (discrete case)
F(x) = P(X <= x)
Practice Yourself
Is P(X=1)=0.3, P(X=2)=0.7 a valid distribution?
Yes, sums to 1
Is P(X=1)=-0.2, P(X=2)=1.2 valid?
No, probabilities cannot be negative
For P(X=1)=0.4, P(X=2)=0.6, find F(1).
0.4