Instructor: Nikhil MuralidharOctober 26, 2022Due Date: November 11, 2022CS 556-B: Mathematical Foundations of Machine LearningHomework 3: Probability & Calculus (100 points)Note: All solutions methods must be fully explained. All solutions must be compiled either using Microsoft Wordor LaTeX and submitted as a PDF file. Hand-written assignment solutions will NOT be accepted.Probability (60 points)1. (20 points) Jane picks apples. The weight of the apples picked by Jane are normally distributed with a meanof 160g and a standard deviation of 50g. Jane does not like apples that weigh less than or equal to 95g.(a) (10 points) What is the percentage of apples that Jane picks and does not like?Hint:Employ Z-scoresand the z-table to calculate the required percentage.(b) (10 points) Letq3represent the minimum weight for the upper quartile (i.e., top 25%) of weight of applesliked by Jane. What is the probability that the weight of a randomly selected apple picked by Jane isgreater than or equal toq3.Hint:Consider the probability that Janelikesan apple AND that the weightof the apple lies in theupper quartile(i.e., weight≥q3)2. (10 points) A bag contains five types of balls colored red, blue, green, yellow, orange. There are 10 each of red,blue, green and yellow balls and there are 3 orange balls. We draw a ball from the bag, if the ball drawn iscolored green or yellow we get$0 payoff. If it is blue the we get a$200 payoff. If the ball drawn is red, we geta$300 payoff and$500 if it is orange.(a) (5 points) What is our expected payoff of this experiment?Hint:Calculate expected value.(b) (5 points) What is the standard deviation and variance?Hint:σ2(X) =E[X2]−(E[X])23. (10 points) Harry is 24 years old and plans to get his driver’s license. The written knowledge test Harry hasto give before the driving test is multiple choice where every question hasyoptions. There are no negativemarks for wrong answers and hence Harry chooses to guess an answer if he doesn’t know the correct answer tothe question. Letxbe the probability that he knows the answer. The probability where Harry guesses at theanswer and it will be correct is1y. What is the probability that Harry knows the correct answer to a problemgiven that he has answered that problem correctly. (Assume that Harry does not leave any questions on thetest unanswered. Also assume that for any question to which harry knows the answer, his answers will alwaysbe correct.)Hint:Employ Bayes rule.4. (10 points) It is estimated that 50% of emails are spam emails. Some software has been applied to filter thesespam emails before they reach your inbox. A certain brand of software claims that it can detect 99% of spamemails, and the probability for a false positive (a non-spam email detected as spam) is 5%. Now if an email isdetected as spam, then what is the probability that it is in fact a non-spam email?5. (10 points) In a study, doctors were asked what the odds of breast cancer would be in a woman, who was initiallythought to have a 1% risk of cancer but who ended up with a positive mammogram result . A mammogramaccurately classifies about 80% of cancerous tumors and 90% of benign tumors.What are the chances thispatient has cancer?Differential Calculus (40 points)

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