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 r < 60 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 supplier 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, if r < 60 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 supplier vou chose in part (b), compute the probability that the total cost does not exceed $2100.
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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.
![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)"](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7b0a5411-bdb0-4804-8919-e2df25fb5bff%2F090aadeb-312a-4c14-a004-c212d6ecbd67%2Fgt372nl_processed.jpeg&w=3840&q=75)
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)"
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