In Exercises 7–10, let H be the hyperplane through the listed points.(a) Find a vector n that is normal to the hyperplane.(b) Find a linear functional f and a real number d such that H = [f : d].

Week 12 notes th March 30 Notes Monte Carol Simulation Use when decisions are made with random events Decision Tree: only random events with discrete outcomes o Discrete outcome=countable or limited number of outcomes Monte Carlo Simulation: random events w/ discrete and continuous outcomes o Continuous outcome=every possible outcome in a range o Can use parametric and structural sensitivity easily =RAND(x) function o Generates random number between 0 and 0.99999… o Re-calculates by pressing F9 (on PC) o Every number is equally likely (uniform distribution) o If make a histogram of 1000 trials of RAND fxn, all bars should be equal height Uniform distribution outside 0-1 o Ex. Get random number between 10 and 25: 10+(25-10) x RAND() 10= lower limit (25-10)=range Why at least 10 (lower limit) and rand can be anywhere between 0-1 so multiply by range to put outcome between 10-25 o Get random integer values between 60 and 75 RANDBETWEEN(60,75) =RANDBETWEEN gives integer values only Simulating 2 outcome discrete distributions o Ex. Simulate event that may happen with probability of 0.4 =IF(RAND()<0.4, 1, 0) Why Rand will give number between 0-1 0-0.4 0.4-1 Hit flop 40% 60% Binary 1=hit and 0=flop Random Number from a Normal Distribution Normal Distribution 3 2 1 0 0 0.2 0.4 0.6 0.8 1 1.2 Ex. Mean=100, SD=20 NORM.INV(RAND, 100, 20)=will give number on x-axis normal distribution probability will create a norm distribution histogram use rand fxn to replace probability Baby Lee’s Doughnuts Baby Lee’s Doughnuts stocks exactly 33 super-duper doughnuts every day. Each super-duper doughnut sells for $2. Unsold doughnuts are eaten by Baby Lee at the end of the day. Daily demand for super-duper doughnuts is a random event and is given by the following probability distribution. Develop a simulation model to determine the mean and standard deviation of daily revenue obtained from the sale of the super-duper doughnuts. Decision tree 0.4 32 o 0.3 33 0.3 34 Make range from 0-1 with 3 possible outcomes3 sub-ranges Boundary of ranges correspond with probabilities of each value 32 33 34 0 0.4 0.7 0.4 0.7 1 40% 30% 30% In excel B C D E 2 Demand 32 33 34 3 Probability 0.4 0.3 0.3 Simulation Calculations Demand: =IF(B10250")/1000 o Give percent chance that of donations being over $250 Percentage changes every time but stays in a certain range Increase number of trialsnarrow range of answers