Homework Sessions 5.1, 5.2 — Joint, Marginal & Conditional Distributions (4×3)

X = minutes in store (T1=5–10, T2=10–20, T3=20–40, T4=40–60).
Y = dollars spent (A=<10, B=10–30, C=30–80). Midpoints: Xmid = [7.5, 15, 30, 50], Ymid = [5, 20, 55].

1) Bins, Counts, Midpoints

Observed Counts (n=—)

Y \\ X T1
5–10		 T2
10–20		 T3
20–40		 T4
40–60		 Row Sum
Col Sum — — — — —

Joint probability table

Y \\ X T1T2T3T4Row Prob
Col Prob — — — — —
Rounding: (counts are fixed)

2) Computed Quantities (with steps)

Marginals

f_X(T1..T4) = [ — ]
f_Y(A..C) = [ — ]
Show how we get marginals

Expectations via midpoints

E[X] = [ 7.5, 15, 30, 50 ] · [ — ] = — minutes
E[Y] = [ 5, 20, 55 ] · [ — ] = $—

2b) Copy-Ready Summary

f_X(T1..T4) = [ — ]
f_Y(A..C) = [ — ]

3) Homework Questions (6)

  1. Q1. Compute the joint probability f(T3, B).
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    2. 
      
  2. Q2. Compute the marginal f_X(T3).
        
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  3. Q3. Compute P(B | T3).
        
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  4. Q4. Compute P(X ≥ 20) (i.e., T3 or T4).
        
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  5. Q5. Using midpoints, compute E[X] and E[Y] (show dot-product).
        
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  6. Q6. Independence check: is f(T3,B) ≈ f_X(T3)·f_Y(B)?
        
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4) Quick Conditional Snapshots

    

      P(B | T3) = — • P(T3 | B) = —