Homework Sessions 5.1, 5.2 — Joint & Marginal & Conditional Distributions (Lotte Market, 4×3)
Variables: X = minutes in store, Y = dollars spent. 4 time bins for X, 3 money bins for Y. We use midpoints only for E[X], E[Y] (no covariance/correlation in this HW).
1) Bins, Counts, Midpoints
Categories
          X (minutes): T1=5–10, T2=10–20, T3=20–40, T4=40–60
Y ($): A=<10, B=10–30, C=30–80
        Y ($): A=<10, B=10–30, C=30–80
Midpoints for expectations: Xmid = {7.5, 15, 30, 50}, Ymid = {5, 20, 55}.
Observed Counts (n=—)
| Y \\ X | T1 5–10 | T2 10–20 | T3 20–40 | T4 40–60 | Row Sum | 
|---|---|---|---|---|---|
| Col Sum | — | — | — | — | — | 
Joint PMF pij = countij / n. Marginals: fX(Tj) = column sum / n; fY(Ai) = row sum / n.
2) Computed Quantities
Marginals & Expectations
          fX(T1..T4) = [—]
fY(A..C) = [—]
        fY(A..C) = [—]
          E[X] = — minutes    •   
          E[Y] = $—
        
      Useful Conditionals
          P(Y=B | X=T3) = —    •   
          P(X=T3 | Y=B) = —
        
      3) Homework Questions (6)
- Q1. Compute the joint probability fXY(X=T3, Y=B).
- Q2. Compute the marginal fX(T3).
- Q3. Compute P(Y=B | X=T3).
- Q4. Compute P(X ≥ 20) (i.e., T3 or T4).
- Q5. Using midpoints, compute E[X] and E[Y].
- Q6. Independence check using one cell: is f(T3,B) ≈ fX(T3)·fY(B)?