Question

(a) Find (22,8,P) and nonempty sets (An)>1 C 8 such that P (liminf An) <liminf (P(Ar)) < limsup(P (A)) <P (limsup An). (b) Given A, B,(A.)21,(B.).>1 C , either prove the following statements or find counterexamples to them. i. limsupA, U limsupB = limsupA, UB.. ii. liminf Anu liminfB = liminfAUB iii. limsupA, nlimsupB. = limsupA, B. iv. liminfa, nliminfB = limin A, B, (e) Prove that probability spaces have the property of continuity from above. (We proved continuity from below in class). (d) Let A = {ACN:|A|< <0 or Ā< }. In homework 3, you proved that if = N thenA is a ſo if A field but not a o-field. For A € A, define the set function S (A) = 3 i. If2 = No, show that S is finitely additive but not a-additive. ii. If 12 = N1, show that S is a-additive. (e) (OPTIONAL) Given f.() : R R define S = {R:fn () + f (x)}. Prove that S = Umilimsup : f(x)-f(x) > }.

Answer 1

Solution The probability of customers generating revenue poj is given by Pij = S:+R ESAR Here P.; = probability of customers generating revenue Si = store size of Shop location ; Roi = annual net revenue of the customer block now to store gr

No. of Avg Furniture Expenses Annual Furniture Households per Household Expense per Block 730 180 131400 1130 125 141250 1035 289800 635 222250 12000 5250 125 7500 470 54050 305 27450 1755 265 465075 900 215 193500 290 370 107300 Total 160 105 Area/Expense (improtance factor) Store A Store B Store x Store Y Store z 2081640348913877602326 20816403489 2775520465213877602326 22702694592 15135129728 2270269459230270259456 15135129728 53778578234 35852385489 53778578234 71704770978 35852385489 39111221754 26074147836 39111221754 52148295672 26074147836 1177900902 785267267.9 1177900902 1570534536 785267267.9 436799792.3 291199861.5 436799792. 3 582399723291199861.5 670138817.7 446759211.8 670138817.7 893518423.6 446759211.8 7168793937 4779195958 7168793937 9558391916 4779195958 3179377362 2119584908 3179377362 4239169816 2119584908 94867805113 63245203409 94867805113 126490406817 63245203409 33121379919 22080919946 33121379919 44161839892 22080919946 16323411366 10882274244 16323411366 21764548488 10882274244 293354505277 195569670185 293354505277 391139340370 195569670185

Census Block No Store A 1 7.10% - 2 7.74% 18.33% 13.33% 0.40% 0.15% 0.23% 8 2.44% 1.08% 10 32.34% 11 11.29% 125.56% Probability of Revenue Generation Store B Store X 7.10% 7.10% 7.74% 7.74% 18.33% 18.33% 13.33% 13.33% 0.40% 0.40% 0.15% 0.15% 0.23% 0.23% 2.44% 2.44% 1.08% 1.08% 32.34% 32.34% 11.29% 11.29% 5.56% 5.56% Store Y 7.10% 7.74% 18.33% 13.33% 0.40% 0.15% 0.23% 2.44% 1.08% 32.34% 11.29% 5.56% Store z 7.10% 7.74% 18.33% 13.33% 0.40% 0.15% 0.23% 2.44% 1.08% 32.34% 11.29% 5.56%
3,4,10 and 11 blocks generate most revenue for all the 5 stores