Question

TomCo Outlet runs a promotion. All customers who visit the store between noon and 1pm on Saturday will receive a gift certificate worth a random amount that is Discrete Uniform on $1,..., $10}. TomCo Outlet managers expect the number of customers to visit the store during the promotion and receive a gift certificate is Discrete Uniform on {51, ..., 100}. What are the expected value and variance of total payouts on the promotional gift certificates?

Please explain the steps, thank you!

Answer 1

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Let Xi be the amount g gift certificate given to ith contumer, as we know X; ~ Uniform {1,2 ... 103 let N number of person which is also uniform receive a gift certificate or uniforms & 51, n. 100% Then the total pay out is given by How the need to calculate E(S) and var (8) (i) E(S) : ECECSIN)) 1. It is well know proved in second last pages of answes se ElS)? E ( E(Xi) +E(X2) -- -- EkN)) Mean oo Expectation of Xi for every i is pet 866) z EXU - - - EKM = EX) , Now some so E(S) z E(NEX)) = E(N) E(X) Xu Dnipoon 40, 2, -- 105 So Ex) = 10!"-5.5 No Unifor uso, ... 100g so E(N) 2 S11100 FS-5- because we know qd yn unig £a, b} E(X) = a + b so E(S) z 5.5x755 z 415.25

(ii) Val(s) = E [val (SIN)] + Vas [E (SIN)] This pooof is also given in last pages/ Vas is on} = Var (xi) - Valltid + Vare) --- Var Kn) in val(x) ECSIN) 2 N E(X) Van () - E ( N. val(x)) + val [Entor N.EX)] - E(N). var(x) + V (N).(E63)? (3-9 + 1-1 (10-1 + 1) - 1 12 12 Var (X) = (3-atee - Z 99, 99 iz 9.25 Vax (N 50-) (100-51 +1)?, = 12 .502.208.25- - . 208.25 12 12 Val(s): 75.5 * 8.25 + 208.25-x(5.5)2 = 622.875 +6299.5625 = 6922.4375

this - There are two Theorems which you need to mad solve 0 If x audy are random variables than ECY) = E[ELYIX)] Proof :- E CE LYıx)] = [e14\x) fax(2) dz = S(Sy flyle) dy) fxle) da = (Syfley) andy = E(N) hence proved (2 9j X and Y are random variable their Var [Y]= E (var (41x)] + val [E (11x] Proof : The variance of a conditional distribution y given x=a is ginar by Var [Y/n]: E [LY - E171233*1*7 = E(4410) LE 1412) Vauly/x] = ELY IX J - (EMIX)). Hence E[vae Cy1x]] - ECE CY?]] - E (ELYIX))} = E(12) - ElE141xje => EQ2), E[valy1x)] + E(E 171x eliga he know that ret var (Y) = E(7²) = (E cys?

Vai (E(Y)x)= ElE (X1XJ)? - (E (EL YIX))) Vai(Y) = E [vae (Y/x)) + E (E (VIX - (E LE (YIX))2 Because we kudw ElE (V1x)) = E(V) so Nar(y) z Erranty 1XJ) + Var (E (VIX)) | Herce Poowed