Nirmala Memorial Foundation College of Commerce & Science Std: S.Y.J.C COMMERCE Subject : Mathematical and Statistics Date :09/08/2014 Practical 11. Probability Distribution 1. The probability mass function of a random variable P( ) = When = -2 , -1 = When =0 = When =1 = 0 otherwise Determine (i) P(X is given by -1), (ii) P(X < 0), (iii) E (X), (iv) V (X) 2. A wheel of fortune at a fair is divided into 5 different colours viz. red, blue, green, yellow and black. The probabilities of the spinner landing in these colours are , , , , respectively. A player wins Rs. 50 if it stops on red, Rs. 30 if It Stops on blue; Rs. 40 if it stops on green, loses Rs. 30 if it stops on yellow. He loses Rs. 10 if it stops on black. Rajesh wants to try his luck. What is his expectation? 3. The probability distribution of a random variable is as follows: X=x -1.5 -0.5 0.5 1.5 P(X = x) 0.05 0.2 0.15 0.25 i) Construct cumulative distribution function F ( ) for X. ii) Obtain P(X 0.5), F (-0.5), F (2), P(X 4). 4. In the following table, c.d.f. of a random variable X is given. X=x -2 -1 0 1 F(x) 0.2 0.5 0.65 0.9 i) Find p.m.f. of X. ii) Compute P(0 X 2), P (X 1), P(X 2/ X 0) 2.5 0.35 2 1 Nirmala Memorial Foundation College of Commerce & Science Std: S.Y.J.C COMMERCE Subject : Mathematical and Statistics Date :10/08/2014 Practical 12. Binomial Distribution 1. The center for disease control has determined that when a person is vaccinated, the probability that the person develops immunity to a virus is 0.8. if eight people are given vaccine, find the probability that, (i) None will develop immunity (ii) Exactly four will develop immunity. (iii) All will develop immunity. 2. The probability that a person who undergoes certain surgery will recover is 0.7. Find the probability that of the six patients who undergo the surgery, (i) None will recover. (ii) All will recover. (iii) Half of them will not recover. (iv) At least half of them will recover. 3. A biased coin for which head is twice as likely as tail is tossed five times. Find the probability of getting. (i) No head (ii) At least one head (iii) Majority of heads. 4. The chance that any of the 5 telephone lines is busy at an instant is 0.01 (i) What is the probability that all the lines are busy? (ii) What is the probability that not more than 3 lines are busy? 5. In 800 families with three children, how many families are expected to have (i) All girls (ii) At least 1 boy (iii) Exact one boy, assuming equal chance of boys and girls. Nirmala Memorial Foundation College of Commerce & Science Std: S.Y.J.C COMMERCE Subject : Mathematical and Statistics Date :12/08/2014 Practical 13. Poisson Distribution 1. For a Poisson distribution with = 0.7, find (i) P(X = 2), (ii) P(X 2), given that e-0.7 = 0.497 2. The probability that an individual will have a reaction, after a particular drug is injected, is 0.001. If 2000 individuals are given the injection, find the probability that more than two have a reaction.(Give that e-2 = 0.135) 3. It is observed that the average number of phone calls per minute coming into switchboard of a company is 3. Find the probability that during a particular minute, there will be (i) no phone calls (ii) exact 2 phone calls (iii) at least 3 phone calls. (Given that e-3 = 0.0498) 4. A car hire firm has two cars which it hires out day by day. The number of demands for a car on each day is distributed as Poisson variate with mean 1.5. Calculate the proportion of days on which (i) neither car is used, and (ii) at least one car is used (e-1.5 = 0.2231) 5. If the variance of the Poisson distribution is 2, find the distribution for r = 1, 2, 3, 4, and 5. (use e-2 = 0.1353) Nirmala Memorial Foundation College of Commerce & Science Std: S.Y.J.C COMMERCE Subject : Mathematical and Statistics Date :16/08/2014 Practical 14. Assignment Problems 1. A company has to assign four workers A,B,C and D to four jobs W, X, Y, and Z the cost matrix is given below : Jobs (cost in Rs.) W X Y Z A 1000 1200 400 900 B 600 500 300 800 Workers C 200 300 400 500 D 600 700 300 1000 Obtain an optimum assignment schedule and total cost required. 2. Four salesmen are to be assigned to four districts. Estimates of the sale revenue in hundred of rupees for each sale are as follows. District Salesman A B C D 1 320 350 400 280 2 400 250 300 220 3 420 270 340 300 4 250 390 410 350 Give the assignment pattern that will maximize the sale revenue. 3. Solve the following unbalanced assignment problem of minimizing total time for doing all the jobs. Job Operator 1 2 3 4 5 1 6 2 5 2 6 2 2 5 8 7 7 3 7 8 6 9 8 4 6 2 3 4 5 5 9 3 8 9 7 6 4 7 4 6 8 Suggest optimum solution to the following assignment problem. Services Time (in hrs) Salesman Counter A B C D W 41 72 39 52 X 22 29 49 65 Y 27 39 60 51 Z 45 50 48 52 4. Nirmala Memorial Foundation College of Commerce & Science Std: S.Y.J.C COMMERCE 1. 2. 3. 4. 5. Subject : Mathematical and Statistics Date :23/08/2014 Practical 15. Sequencing Five jobs are to be processed. Each job must go through two machines A and B in order AB. The time required for processing (in minutes) these jobs is given in the following table. Job 1 2 3 4 5 A 6 2 10 4 11 B 3 7 8 9 5 Determine a sequence for these jobs which will minimize total required. Also find the idle time for each machine. Find the optimal sequence which minimizes total required for completing five jobs which have to be processed on each of the machines M1, M2, M3 in that order. The processing time for each Job on machine is given below. Also find the total elapsed time and idle time for each machine. Jobs 1 2 3 4 5 M1 6 7 5 11 5 M2 4 3 2 5 1 M3 3 8 7 4 9 Find the sequence that minimized the total time required in performing the following jobs on three machines in the order X Y Z. Processing time (in minutes) Jobs Machine 1 2 3 4 5 6 8 3 7 2 5 1 X 3 4 5 2 1 6 Y 8 7 6 9 10 9 Z The time (in hours) required to perfume the printing and binding operations in that order for each book is given in the following table : Book I II III III IV Printing Machine M1 3 7 4 5 7 Printing Machine M2 6 2 7 3 4 Find the sequence that minimizes the total elapsed time (in hours) to complete the work. Also find the minimum elapsed time T and times for two machines. Five jobs are to be processed on three machines X, Y, Z in the order YXZ. The time schedule for the entire task is as follows: Machines Jobs P Q R S T X 5 6 2 3 4 Y 4 9 8 6 5 Z 8 10 6 7 11 Determine the optimal sequence, state the total elapsed time and optain the idle times for machines, X, Y, Z. Nirmala Memorial Foundation College of Commerce & Science Std: S.Y.J.C COMMERCE Subject : Mathematical and Statistics Date :30/08/2014 Practical 16. Applications of Derivatives ( ) 1. The cost of production of assembly cost 625 + 2 bulbs consists of labour cost Rs. 3 , and packing charges are . Find (i) Total cost of producing bulbs. (ii)Average cost of producing 10 bulbs. (iii) Find values of decreasing. for which average cost of production is 2. The price of a good is P; demand (D) and supply (S) are given as D = and S = P2. Find (i) marginal demand and marginal supply at the equilibrium price. (ii) Are demand and supply increasing at equilibrium price? 3. The demand function is given by P = 18 Find (i) Total revenue function. (ii)Marginal revenue at D = 2. (iii) For what value of demand the revenue is increasing? 4. 5. The demand y for a commodity when its price is x is given by y = Find (i) elasticity of demand for P = 3. (ii) Marginal revenue when price is 3 units. The demand curve is given as P = (10 - 2D) (5 – D2). Find elasticity of demand for D = 1, D = 2. Also find elasticity of demand when marginal revenue is zero. Nirmala Memorial Foundation College of Commerce & Science Std: S.Y.J.C COMMERCE Subject : Mathematical and Statistics Date :06/09/2014 Practical 17. Applications of Derivatives ( ) 1. The total cost of producing articles depends upon labour cost and processing cost. The labour cost 9 + 16 and processing cost is - 2 2.The total revenue for these articles is R = ( 2 - 24 ). Find the number of articles , to be produced so that (i) revenue (ii) Total profit is maximum. Find profit at this value of for which marginal cost is minimum. 2. 3. 4. Find x for which (i) the total cost function( c) is minimum, where C = 3 - 24 2 + 189 (ii) Marginal cost function is minimum. If the demand function is D = 4 3 - 72 2 + 420 + 75, for what price demand is (i) Maximum, find that demand. (ii) Minimum; find minimum demand. the A firm produces tones of output at a total cost C = - 4 2 + 64 + 47. Find minimum marginal cost for output and average cost at that output. Nirmala Memorial Foundation College of Commerce & Science Std: S.Y.J.C COMMERCE Subject : Mathematical and Statistics Date :13/09/2014 Practical 18. Applications of Definite Integration 1. Find the consumption function (E) and value of APC, MPS if MPC = (0.0002) + (0.01) and consumption is 2000 when income is 3000. 2. The marginal revenue and the marginal cost for an output of a commodity is given as RM = 8 - 4 + 6 2 and CM = 3 + 2 . For fixed cost zero and if revenue is 10 for unit output, find the profit function and the profit when output is 4. 3. Find the profit function when marginal cost function for an output is given as CM = 3 2 - 2 + 9 and fixed cost is 200 and marginal revenue functionis RM = 6 2 – 8 where revenue is 80 for = 3. 4. Determine the cost of producing 5000 pens if the marginal cost CM = Also find average cost. + 3. Nirmala Memorial Foundation College of Commerce & Science Std: S.Y.J.C COMMERCE Subject : Mathematical and Statistics Date :20/09/2014 Practical 19. Commission, Brokerage and Discount 1. The list price of articles is 25% above the selling price and cost price is 40% below the list price. Find the percentage rate of discount and percentage profit. 2. Mr. Patel purchased a shop worth Rs. 4, 00,000, through a broker. The broker charged 2 % brokerage. Find the amount the Mr. Patel had to pay in purchasing the shop and also the amount the broker received as brokerage. 3. Find the true discount, banker’s discount and banker’s gain on a bill of 61,000 payable after 6 months at 4 % interest per annum. 4. A bill of Rs. 50,000 legally due on June 13 and another for Rs. 40,000 legally due on August 25 are both discounted by the broker on April 1 st. If the difference between two discounts is Rs. 240, find the rate at which the discount is calculated. 5. If the difference between the true discount and bankers discount on a bill payable 4 months hence at 6 % per annum is Rs. 50, find (i) bill value (ii) true discount (iii) banker’s discount. Nirmala Memorial Foundation College of Commerce & Science Std: S.Y.J.C COMMERCE Subject : Mathematical and Statistics Date :27/09/2014 Practical 20. Insurance and Annuity 1. A person is insured for Rs. 3 lakh for 25years. He dies after 15 years. If the rate of annual premium is Rs. 45 per thousand and the annual bonus is Rs. 40 per thousand, what amount did his nominee receive? 2. A car valued at Rs. 4, 50,000 is insured for Rs. 2, 70, 000. The rate of premium is 2 % less 20%. How much loss does the owner bear including the premium,If the value of the vehicle is reduced to 60% of its original value? Calculate the saving made in the premium. 3. An over draft of Rs. 5, 00,000 is to be paid back in equal; annual installments over a period of 20years. Find the value of the installment, if interest is compounded annually at 14% p.a. (Given (1.14)20 = 13.74349) 4. A company borrows a loan of Rs. 4, 00,950 on condition to reply it with compound interest at 6 % p.a. by annual installments of a Rs. 1, 50,000 each. In how many years will the debit be paid off?