Probability Statistical and Numerical techniques Assignment-1 PDF

Rajiv Gandhi Govt. Engg. College Kangra at Nagrota Bagwan Probability Statistical and Numerical techniques (MAFC-311) Assignment-1 Allotment Date: 07.09.2024 Submission Date: 05.11.2024...

Rajiv Gandhi Govt. Engg. College Kangra at Nagrota Bagwan Probability Statistical and Numerical techniques (MAFC-311) Assignment-1 Allotment Date: 07.09.2024 Submission Date: 05.11.2024 Session 2024-25 M. M.: 08 (Eight) All questions and answers should be presented precisely. Do your own solutions. Copy cases will result in zero marks. Every student will be asked to present any random solution on white board, only then marks will be awarded. First Review Date: 28.09.2024 Second Review Date: 10.10.2024 Third Review Date: 25.10.2024 1. Answer in short: a) If mean of a Poisson distribution is 𝑚, then S.D. of this distribution is….. b) In a Poisson distribution if, 2𝑃(𝑥 = 1) = 𝑃(𝑥 = 2), then man of x is…. c) The odds in favor of an event A are 5:4, then probability of success of A is …. d) Baye’s Theorem states that ………. e) If 𝑝. 𝑑. 𝑓. of 𝑥 𝑖𝑠 𝑓(𝑥) = 𝑘𝑥(1 − 𝑥), 0 < 𝑥 < 1, 𝑡ℎ𝑒𝑛 𝑘 = ⋯ … f) If 𝑉(𝑥) = 2, 𝑡ℎ𝑒𝑛 𝑉(2𝑥 + 1) =……………. g) If the probability of hitting a target by one shot be 𝑝 = 0.8, then the probability that out of ten shots , seven will hit the target is………….. h) Define Exponential Distribution i) Define Binomila distribution. j) Find the value of 𝐸 −1 ∇. k) What is the nth difference of polynomials of degree 𝑛. 3 l) Write Newton’s iterative formula to find the value of √𝑁. [Common all CO’s ] 2. Suppose that two-dimensional continuous random variable (X,Y) has joint pdf given by: 6𝑥 2 𝑦 , 0 < 𝑥 < 1, 0 < 𝑦 < 1 𝑓(𝑥, 𝑦) = { 0 , 𝑒𝑙𝑒𝑠𝑤ℎ𝑒𝑟𝑒 1 1 a) Verify that ∫0 ∫0 𝑓(𝑥, 𝑦)𝑑𝑥 𝑑𝑦 = 1 3 1 b) Find 𝑃 (0 < 𝑋 < 4 , 3 < 𝑌 < 2) , 𝑃(𝑋 + 𝑌 < 1), 𝑃( 𝑋 > 𝑌)𝑎𝑛𝑑 𝑃(𝑋 < 1 / 𝑌 < 2). [CO1] 3. A multiple choice test consists of 8 questions with 3 answers to each question (of which only one is correct). A student answers each question by rolling a balanced die and checking the first answer if he gets 1 or 2, the second answer if he gets 3 or 4 and the third answer if he gets 5 or 6. To get distinction, he student must secure at least 75% of correct answers. If there is no negative marking, what is the probability that the student secures a distinction? [CO1] 4. The mean height of 500 students is 151cm and the standard deviation is 15 cm. Assuming that the heights are normally distributed, find how many students hights lie between 120 and 155 cm. [CO2] 5. A machine produces 16 imperfect articles in a samle of 500. After machine is overhauled, it produces 3 imperfect articles in a batch of 100. Has the machine been improved? [CO2] 6. Two horses A and B were tested according to the time(in seconds) to run a particular race with the following results: Horse A 28 30 32 33 33 29 34 Horse B 29 30 30 24 27 29 --- Test whether you can discriminate between horses? [CO3] 7. A die was thrown 60 times and the following frequency distribution was observed: Faces 1 2 3 4 5 6 f 15 6 4 7 11 17 Test whether the die is unbiased? [CO3] 8. Find the polynomial 𝑓(𝑥) by using Lagrange’s formula and hence find 𝑓(3) for the given data: 𝑥 0 1 2 5 𝑓(𝑥) 2 3 12 147 6 9. Evaluate ∫0 𝑥𝑠𝑒𝑐𝑥 𝑑𝑥 using six intervals by Trapezoidal rule. [CO5] 𝑑𝑦 10. Consider an ordinary differential equation 𝑑𝑥 = 𝑥 2 + 𝑦 2 , 𝑦(1) = 1.2. Find 𝑦(1.05) using the fourth order Runge-Kutta method. [CO5] 11. Using Taylor series method of order four to solve the initial value problem 𝑦′ = (𝑥 − 1 1 1 𝑦)/2, on [0, 3] with y(0) = 1. Compare solutions for ℎ = 1, , and 8. [CO5] 2 4 12. Using, Newton’s Raphson method, find the real root of the equation 3𝑥 = 𝑐𝑜𝑠𝑥 + 1. Also, evaluate the value of √5 by using Newton’s method. [CO4] 13. Solve the system of equations 10𝑥 − 7𝑦 + 3𝑧 + 5𝑢 = 6, −6𝑥 + 8𝑦 − 𝑧 − 4𝑢 = 5, 3𝑥 + 𝑦 + 4𝑧 + 11𝑢 = 2 , 5𝑥 − 9𝑦 − 2𝑧 + 4𝑢 = 7 by using Gauss elimination method. [CO4]