P(d = k) = {1/10 if k = 5 3/10 if k = 6 2/5 if k = 7 1/10 if k = 8 1/10 if k = 8
ID: 3201374 • Letter: P
Question
P(d = k) = {1/10 if k = 5 3/10 if k = 6 2/5 if k = 7 1/10 if k = 8 1/10 if k = 8 1/10 if k = 9 0 otherwise. David buys fruits and vegetables wholesale and retails them at Davids Produce on La Vista Road. One of the more difficult decisions is the amount of bananas to buy. Let us make some simplifying assumptions, and assume that David purchases bananas once a week at 10 cents per pound and retails them at 25 cents per pound during the week. Bananas that are more than a week old are too ripe and are sold for 5 cents per pound. Suppose the demand for the good bananas follows the same distribution as D given in the first question. What is the expected total cost of David in a week if he buys 8 pounds of banana? Now assume that the demand for the good bananas is uniformly distributed between 5 and 10. What is the expected cost of David in a week if he buys 7 pounds of banana? Find the expected cost if David's demand for the good bananas follows an exponential distribution with mean 6 and if he buys 6 pounds of banana.Explanation / Answer
Given,
Cost price=10
Selling price=25
Profit per unit=25-10=15
Salvage value=5
loss for old bananas=10-5=5
Total expected cost=Profit lost in sales+loss made in leftover bananas
A.3)Total expected cost for P(D=8)=profit lost in P(D=9) + [loss made in P(D=5)+P(D=6)+P(D=7)]
Total expected cost for P(D=8)= 1/10*(25-10)*1+1/10*(10-5)*3+3/10*(10-5)*2+2/5*(10-5)*1
Total expected cost for P(D=8)=1.5+1.5+3+2=8 dollars
hence, total expected cost for buying 8 pounds of bananas is 8 dollars
A.4) In case of uniformly distributed from 5 to 10, probabilities from P(D=5) to P(D=10) will be 1/6
Total expected cost for P(D=7)=Profit lost in sales+loss made in leftover bananas
Total expected cost for P(D=7)=[profit lost in P(D=8)+P(D=9)+P(D=10)] + [loss made in P(D=5)+P(D=6)]
Total expected cost for P(D=7)=1/6*[1*15+2*15+3*15]+1/6*[2*5+1*5]
Total expected cost for P(D=7)=17.5 dollars
hence, total expected cost for buying 7 pounds of bananas in a uniform distribution from 5 to 10 is 17.5 dollars
A.5) At mean we achieve optimal point and profit maximization stage
hence, expected cost for the exponential distribution is zero
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