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A retailer is trying to decide between two suppliers to purchase 100 units a product. Supplier 1 offers a price of $25 per product but has a buy back guarantee for any unsold items at a price of $10 per item. Supplier two has a cheaper price of $22 per item but offers only $5 as the buyback price. Suppose that the retailer estimates the demand for the item to follow the cumulative distribution function below. (0, 0.4, if 60 100 (a) Write the net cost of purchasing the product after the buyback as a function of demand X for each supplier. (b) If the retailer's objective is to minimize the expected net cost, which supplier should they choose? What would be the resulting cost? (c) For the sunplie" vou chose in part (b), compute the probability that the total cost does not exceed $2100.

A retailer is trying to decide between two suppliers to purchase 100 units a product. Supplier 1 offers a price of $25 per product but has a buy back guarantee for any unsold items at a price of $10 per item. Supplier two has a cheaper price of $22 per item but offers only $5 as the buyback price. Suppose that the retailer estimates the demand for the item to follow the cumulative distribution function below. (0, 0.4, if 60 100 (a) Write the net cost of purchasing the product after the buyback as a function of demand X for each supplier. (b) If the retailer's objective is to minimize the expected net cost, which supplier should they choose? What would be the resulting cost? (c) For the sunplie" vou chose in part (b), compute the probability that the total cost does not exceed $2100.

Functions and Change: A Modeling Approach to College Algebra (MindTap Course List)
Functions and Change: A Modeling Approach to College Algebra (MindTap Course List)
6th Edition
ISBN:9781337111348
Author:Bruce Crauder, Benny Evans, Alan Noell
Publisher:Bruce Crauder, Benny Evans, Alan Noell
Chapter2: Graphical And Tabular Analysis
Section2.1: Tables And Trends
Problem 2TU: Suppose one canoe rents for 40,and2 is taken off the price for each additional canoe rented by a...
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Attached formula sheet! Please AMSWER ASAP
Axioms of Probability Also Note: 1. P(S)=1 For any two events A and B, 2. For any event E, 0< P(E)S1 P(A) = P(AN B) + P(An B') 3. For any two mutually exclusive events, and P(EUF) = P(E) + P(F) P(An B) = P(A|B)P(B). Events A and B are independent if: Addition Rule P(EUF) = P(E) + P(F) - P(En F) P(A|B) = P(A) or Conditional Probability P(An B) = P(A)P(B). P(B|A) = PANB %3! P(A) Bayes' Theorem: Total Probability Rule P(A|B)P(B) P(A) = P(A|B)P(B) + P(A|B')P(B') P(B|A) = P(A|B)P(B) + P(A|B')P(B') Similarly. Similarly, P(A) =P(A|E)P(E) + P(A|E2)P(E2)+ P(B|E1)P(E) P(Ei|B) = ...+ P(A|E)P(E) P(B\E,)P(E,)+ P(B\E,)P(E») + ·…· + P(B\E4)P(E Probability Mass and Density Functions Cumulative Distribution Function • F(r) = P(X Sz) If X is a discrete r.v: lim,- F(x) = 0 • P(X = r) = f(r) • lim,- F(r) = 1 Es(z) = 1 (total probability) • F(r) = Eus S(z) if X is a discrete r.v. • P(a <X sb) = F(b) – F(a) Expected Value and Variance Expected Value of a Function of a RV • E[X] = E,rf(x) if X is a discrete r.v. • E'h(X)) = E, h(z)f(z) if X is a discrete r.v. • E[X] = [ rf(r)dr if X is a continuous r.v. • E'h(X)] = * h(r)f(x)dr if X is a continu- %3D ous r.v • Var(X) = E[X2] – E[X}² • EļaX + b] = aE[X] +6 • Var(X) = E[(x - EX))") • Var(aX +b) = a?Var(X) Common Discrete Distributions •X - Bernoulli(p). if r = 1; f(r) = 11-p if z=0 E[X] = p. Var(X) = p(1 – p). • X- Geometric(p), S(x) = (1- p)-'p, z€ {1,2,.), E[X] = Var(X) = . Geometric Series: Eo -. for 0 <q<1 • X- Binomial(n, p), S(r) = (")(1 – p)"-"p", r€ (0, 1,., n}, EX] = np, Var(X) = np(1 – p). %3D • X- Negative Binomial(r, p), S(2) = ()(1- p)*-,z€ {r,r+1,.) E[X] = 5, Var(X) = ", • X - Hypergeometric(n, M, N), fa)=쁘, 티지] = "불, Var(X) 3극"북 (1-부). • X- Poisson(AM), S(1) = , z € {0, 1,.}, E[X] = tt, Var(X) = t. A (M)"
Transcribed Image Text:Axioms of Probability Also Note: 1. P(S)=1 For any two events A and B, 2. For any event E, 0< P(E)S1 P(A) = P(AN B) + P(An B') 3. For any two mutually exclusive events, and P(EUF) = P(E) + P(F) P(An B) = P(A|B)P(B). Events A and B are independent if: Addition Rule P(EUF) = P(E) + P(F) - P(En F) P(A|B) = P(A) or Conditional Probability P(An B) = P(A)P(B). P(B|A) = PANB %3! P(A) Bayes' Theorem: Total Probability Rule P(A|B)P(B) P(A) = P(A|B)P(B) + P(A|B')P(B') P(B|A) = P(A|B)P(B) + P(A|B')P(B') Similarly. Similarly, P(A) =P(A|E)P(E) + P(A|E2)P(E2)+ P(B|E1)P(E) P(Ei|B) = ...+ P(A|E)P(E) P(B\E,)P(E,)+ P(B\E,)P(E») + ·…· + P(B\E4)P(E Probability Mass and Density Functions Cumulative Distribution Function • F(r) = P(X Sz) If X is a discrete r.v: lim,- F(x) = 0 • P(X = r) = f(r) • lim,- F(r) = 1 Es(z) = 1 (total probability) • F(r) = Eus S(z) if X is a discrete r.v. • P(a <X sb) = F(b) – F(a) Expected Value and Variance Expected Value of a Function of a RV • E[X] = E,rf(x) if X is a discrete r.v. • E'h(X)) = E, h(z)f(z) if X is a discrete r.v. • E[X] = [ rf(r)dr if X is a continuous r.v. • E'h(X)] = * h(r)f(x)dr if X is a continu- %3D ous r.v • Var(X) = E[X2] – E[X}² • EļaX + b] = aE[X] +6 • Var(X) = E[(x - EX))") • Var(aX +b) = a?Var(X) Common Discrete Distributions •X - Bernoulli(p). if r = 1; f(r) = 11-p if z=0 E[X] = p. Var(X) = p(1 – p). • X- Geometric(p), S(x) = (1- p)-'p, z€ {1,2,.), E[X] = Var(X) = . Geometric Series: Eo -. for 0 <q<1 • X- Binomial(n, p), S(r) = (")(1 – p)"-"p", r€ (0, 1,., n}, EX] = np, Var(X) = np(1 – p). %3D • X- Negative Binomial(r, p), S(2) = ()(1- p)*-,z€ {r,r+1,.) E[X] = 5, Var(X) = ", • X - Hypergeometric(n, M, N), fa)=쁘, 티지] = "불, Var(X) 3극"북 (1-부). • X- Poisson(AM), S(1) = , z € {0, 1,.}, E[X] = tt, Var(X) = t. A (M)"
A retailer is trying to decide between two suppliers to purchase 100 units a product. Supplier 1 offers a price of $25 per product but has a buy back guarantee for any unsold items at a price of $10 per item. Supplier two has a cheaper price of $22 per item but offers only $5 as the buyback price. Suppose that the retailer estimates the demand for the item to follow the cumulative distribution function below. 0, if a < 60 0.4, if 60 < x < 80 F(x) = 0.6, if 80 < x < 100 1, if a > 100 (a) Write the net cost of purchasing the product after the buyback as a function of demand X for each supplier. (b) If the retailer's objective is to minimize the expected net cost, which supplier should they choose? What would be the resulting cost? (c) For the sunplier vou chose in part (b), compute the probability that the total cost does not exceed $2100.
Transcribed Image Text:A retailer is trying to decide between two suppliers to purchase 100 units a product. Supplier 1 offers a price of $25 per product but has a buy back guarantee for any unsold items at a price of $10 per item. Supplier two has a cheaper price of $22 per item but offers only $5 as the buyback price. Suppose that the retailer estimates the demand for the item to follow the cumulative distribution function below. 0, if a < 60 0.4, if 60 < x < 80 F(x) = 0.6, if 80 < x < 100 1, if a > 100 (a) Write the net cost of purchasing the product after the buyback as a function of demand X for each supplier. (b) If the retailer's objective is to minimize the expected net cost, which supplier should they choose? What would be the resulting cost? (c) For the sunplier vou chose in part (b), compute the probability that the total cost does not exceed $2100.
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