Problem 1 (Wooldridge B.10): Suppose that at a large university, college grade
point averages, 𝐺𝑃𝐴, and SAT score, 𝑆𝐴𝑇, are related by E[𝐺𝑃𝐴|𝑆𝐴𝑇] = 0.70 +
0.002𝑆𝐴𝑇.
(i)
(ii)
(iii)
Find the expected 𝐺𝑃𝐴 when 𝑆𝐴𝑇 = 800. Find E[𝐺𝑃𝐴|𝑆𝐴𝑇 = 1400].
Comment on the difference
E[GPA|SAT] = 0.70 + 0.002(800)
= 2.3
E[GPA|SAT] = 0.70 + 0.002(1400)
= 3.5
Comment: The expected GPA of a greater SAT score is much better
compared to the expected GPA of a lower SAT score.
If the average 𝑆𝐴𝑇 in the university is 1100, what is the average 𝐺𝑃𝐴?
E[GPA|SAT] = 0.70 + 0.002(1100)
= 2.9
If a student’s SAT score is 1100, does this mean he or she will have the
GPA found in (ii)? Explain
No. he or she will get a GPA above or below 2.9
Problem 2: Let {(𝑥𝑖 , 𝑦𝑖) : 𝑖 = 1, 2, ..., 𝑛} be a sample. 𝐿𝑒𝑡 𝑥 =
𝑛
𝑛𝑖=1 𝑥𝑖
(i)
𝑎𝑛𝑑 𝑦 = 𝑛
1 ∑𝑛𝑖=1 𝑦𝑖
Show that
𝑛
𝑛
∑ 𝑥𝑖 − ∑ 𝑥
𝑖
𝑖=1
∑ 𝑥𝑖 − 𝑛𝑥
𝑖=1
𝑛
𝑛
∑ 𝑥𝑖 − ∑ 𝑥𝑖
𝑖𝑖=1
(ii)
Proven.
Using the result in part (i), show that
𝑛
𝑛
∑(𝑥𝑖 − 𝑥 )2 = ∑ 𝑥(𝑥𝑖 − 𝑥)
𝑖=1
𝑖=1
𝐿𝐻𝑆
2
𝑛
∑(𝑥𝑖 − 𝑥)(𝑥𝑖 − 𝑥)
𝑖=1
𝑛
𝑛
∑ 𝑥𝑖2 − 2𝑥 ∑ 𝑥𝑖 − 𝑛𝑥2
𝑖=0
𝑖=0
𝑛𝑥2
𝑛𝑥2
𝑛
𝑛
∑ 𝑥𝑖2 − ∑ 𝑥𝑖 𝑥
𝑖=0
𝑖=0
𝑛
∑ 𝑥𝑖(𝑥𝑖 − 𝑥)
𝑖=0
(iii) Show that
𝑛
𝑛
𝑛
∑(𝑥𝑖 − 𝑥)(𝑦𝑖 − 𝑦) = ∑ 𝑥𝑖(𝑦𝑖 − 𝑦) = ∑ 𝑦𝑖(𝑥𝑖 − 𝑥)
𝑖=0
𝑖=0
LHS
𝑛
∑(𝑥𝑖 − 𝑥)(𝑦𝑖 − 𝑦)
𝑖=0
𝑛
∑(𝑥𝑖𝑦𝑖 − 𝑥𝑖𝑦 − 𝑥𝑦𝑖 − 𝑦𝑖 + 𝑥𝑦)
𝑖=0
𝑛
𝑖=0
∑[(𝑥𝑖(𝑦𝑖 − 𝑦) − 𝑥(𝑦𝑖 − 𝑦)]
𝑖=0
𝑛
∑ 𝑥𝑖(𝑦𝑖 − 𝑦)
𝑖=0
Problem 3: Let 𝑋 and 𝑌 be random variables. Hint: this problem is very similar
to the previous one.
(i)
(ii)
(iii)
Show that E[(𝑋 − E[𝑋])] = 0
LHS
E[(X – E[X])]
E[(X – X)]
E[0]
0
Show that Var(𝑋) = E[𝑋(𝑋 − E[𝑋])]
LHS
Var(𝑋) = 0
RHS
E[𝑋(𝑋 − E[𝑋])]
E[X(X-X)]
E[X(0)]
E[0]
0
LHS = RHS
Show that Cov(𝑋, 𝑌) = E[𝑋(𝑌 − E[𝑌])] = E[(𝑋 − E[𝑋])𝑌]
Cov(X,Y) = 0
E[𝑋(𝑌 − E[𝑌])]
E[X(Y – Y)]
E[X(0)]
E[0]
0
E[(𝑋 − E[𝑋])𝑌]
E[(X – X)Y]
E[(0)Y]
E[0]
0