Uniform Distribution Part 2 Intermediate and Advanced Topics Continuous Probability Distributions Uniform Probability Distribution- Introduction Uniform Probability Distribution- Example

The News Vendor Problem Swell Productions is sponsoring an outdoor conclave for owners of collectible and classic Fords. The concession stand in the T-Bird area will sell clothing such as official Thunderbird racing jerseys. Suppose the probability of jerseys sales quantities is uniformly (and continuously) distributed between 100 and 400. Suppose sales price is $80 per jersey, purchase cost is $40, and unsold jerseys are returned to the manufacturer for $20 per unit. How many Jerseys Swell Production orders? 100 400 Cu: Underage Cost Cu = 80-40 = 40 Co: Overage Cost Q If Co was also 40 40 40 Gray /(White + Gray) Cu/(Cu+Cp) = 40/(40+40) = 0.5 20 40 Co = 40-20 = 20  40/(20+60) =2/3

The News Vendor Problem SL* = Cu/(Cu+Co) SL* = 40/(40+20) = 2/3 (Q-100)/(400-100) = 2/3 Q= 300 100 400 Q

The News Vendor Problem The expected number of participants in a conference is uniformly distributed between 100 and 700. The participants spend one night in the hotel and the cost is paid by the conference. The hotel has offered a rate of $200 per room if a block of rooms is reserved (non-refundable) in advance. The rate in the conference day is $300. All rooms will be single occupied. How many rooms should we reserve in the non-refundable block to minimize our expected total cost. Service level (Probability of demand not exceeding what we have ordered) SL* = Cu/(Cu+Co) Co: Overage cost Co = 200. Cu: Underage cost Cu = 300-200 = 100 100 700

The News Vendor Problem SL* = Cu/(Cu+Co) SL* = 100/(100+200) = 1/3 SL* = (Q-a)/(b-a) = (Q-100)/600 = 1/3 ? 1/600 0.3333 a=100 b=700 Q= 300 B-a=600

A Non-Trivial Problem – Curve, Solver, Data table Profit = (12-x) E(Profit) = 0.2(x-4)(12-x) E(profit) = -0.2x2+3.2x-9.6 x = price offered. Probability of wining = 0.2(x-4)

Estimating the Demand Curve

Estimating the Demand Curve

From Economics to Statistics 160 a= 0, b= 160 f(x) =1/160 µ= (0+160)/2 σ2 = (160-0)2/12 If price is set to P, probability of sale is 1/160(160-P) = 1-P/160 160 P What Price maximizes our revenue? If V= 75, What Price maximizes our profit?

From Uniform Dist. to Binomial Distribution If price is set to P, probability of sale is 1-P/160 Probability of not sale is 1-(1-P/160)= P/160 If 200 people visit this site, what is average sales? What is standard deviation of sales?