LESSON 4.1 · ABOUT 20 MINUTES
Joint probability tables
Describe two random variables together before looking at either alone.
By the end of this lesson
- Validate a joint PMF.
- Compute event probabilities and a joint CDF.
Read a cell
A joint mass p(x,y) is the probability that X=x AND Y=y. Each cell is nonnegative; all cells sum to one. A row or column label is a value, not an event probability.
Combine cells
For an event such as X+Y≤1, enumerate the pairs that satisfy the inequality, then add their masses. Distinct pairs are mutually exclusive.
Joint CDF
F(a,b)=P(X≤a,Y≤b). Sum only the cells meeting BOTH cutoffs. To obtain P(X>a,Y>b), use 1−F_X(a)−F_Y(b)+F(a,b).
F(a,b) = Σ[x≤a, y≤b] p(x,y)
Worked example
For rows X=0,1 and columns Y=0,1, let the table be [[0.1,0.2],[0.3,0.4]]. Find P(X+Y=1).
The qualifying cells are (0,1) and (1,0): 0.2+0.3=0.5. Do not include (1,1), which has sum 2.