If manufactures m pieces on any day the cost in

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If manufactures m pieces on any day the cost (in rupees) in m 2 = 2m. If the number of pieces demanded (n) on any day is less than the production m i.e. of n is less than or equal to m, then all the n pieces demanded are sold and the scale proceeds (in rupees) are 3 n = 3n. And if the number of pieces demanded on any day is greater than the production i.e if n is greater than m, then the maximum supply is limited to m pieces (which is actual production) and thus the sale proceeds (in Rs. ) are 3xm=3m. Hence,

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31 Profit = Sale proceeds Cost = 3n-2m, if n is less than or equal to m = 3n-2m, if n is greater than m Therefore, Profit = 3m-2m, if n is less than or equal to m = m, if n is greater than m. Calculation of net profit or loss for each level of demand assuming the production of 12 pieces N Profit Probability 10 11 12 13 14 15 10 (3 12) 2 = 6 11 (3 12) 2 = 9 12 (3 12) 2 = 12 1 12 = 12 1 12 = 12 1 12 = 12 0.07 0.10 0.23 0.38 0.12 0.10 The profit for 12 pieces (assured to have produced only 12) is calculated on using (X). We notice that for the first three cases (i.e. n=10,11,12) we have n is lesser than or equal to m; and for the remaining three cases (i.e. n=13,14,15) n is greater than m. Expected net profit (X) = Profit x Probability = 6 × 0.07 + 9 × 0.10 + 12 × 0.23 + 12 × 0.38 + 12 × 0.12 + 12 × 0.10 = 0.42 + 0.90 + 2.76 + 5.46 + 5.46 + 1.44 + 1.20 = Rs.11.28 Calculation of net profit for different levels of production based on production based on profit = 3n 2m, if n is less than or equal to m, and profit = m if the n is greater than m, the profits for different levels of demand (n) and production on any day is calculated and is given below. Profit for m Demand (n) Production (m) Probability 10 11 12 13 14 15 10 11 12 13 14 15 10 10 10 10 10 10 8 11 11 11 11 11 6 9 12 12 12 12 4 7 10 13 13 13 2 5 8 11 14 14 0 3 6 9 12 15 0.07 0.10 0.23 0.38 0.12 0.10 Expected net profit (Rs.) 10.00 10.79 11.28 10.08 9.74 8.04
32 Expected net profit calculated as follows: When production (m) =10 10 (0.07 + 0.10 + 0.23 + 0.38+ 0.12+ 0.10) = Rs.10.00 m = 118 0.07 + 11 (0.10+0.23+0.38+0.12+10) = Rs.10.79 m = 126 0.07 + 9 0.10 + 12 (0.23 + 0.38+ 0.12 + 10) = Rs.11.28 m =134 0.07 + 7 0.10 + 10 0.23 + 13 (0.38+0.12+0.10) = Rs.11.08 m =142 0.07 5 0.10 +8 0.23 + 11 0.38 + 14 (0.12+0.10) = Rs.9.74 m = 150.0 0.07 + 3 0.10 + 6 0.23 +9 0.38 + 12 0.12 + 15 0.10 = Rs.8.04 From the expected net profit given is table, we conclude that the maximum expected profit is Rs.11.28 which gets when production (m) is 12. Hence, the production of 12 pieces per day will optimize her (Ravali’s food stall enterprise) expected profit. Mathematical Expectation and Variance The concept, mathematical expectation also called the expected value, occupies an important place is statistical analysis. The expected value of a random variable weighted arithmetic mean of the probabilities of the values that the variable can possibly assume. Robert L. Brite is defined the mathematical expectation as : It is the expected value of outcome in the long run. In other words, it is the sum of each particular value within the set (X) multiplied by the probability. Symbolically, ipi x (X) E n 1 i Variance = (x) 2 - [ (x) ] 2 (x) 2 = x 2 px.

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