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

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) = [—]
E[X] = — minutes    •    E[Y] = $—

Useful Conditionals

P(Y=B | X=T3) = —    •    P(X=T3 | Y=B) = —

3) Homework Questions (6)

  1. Q1. Compute the joint probability fXY(X=T3, Y=B).
  2. Q2. Compute the marginal fX(T3).
  3. Q3. Compute P(Y=B | X=T3).
  4. Q4. Compute P(X ≥ 20) (i.e., T3 or T4).
  5. Q5. Using midpoints, compute E[X] and E[Y].
  6. Q6. Independence check using one cell: is f(T3,B) ≈ fX(T3)·fY(B)?