probability assignment.pdf - Probability Assignment STAT...

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INSTRUCTIONS Presenting your work : You may type out your work in a text editor such as Word or equation editors such as LaTeX or RMarkdown . You may also submit scans/photographs of handwritten work. If you choose this option, ensure your work is clear and easy to read. A combination of typed out and handwritten work is also acceptable. You must present all R commands you used and their outputs. Someone reading your work must be able to take your commands, copy them into R , and obtain identical results. You must clearly indicate your final answer to each of the questions (presenting R output without clearly indicating what number represents your final answer will not be awarded full points). Final numerical answers must be presented with at least 3 significant digits after the decimal point. E.g. 0.000120. Be careful not to round intermediary results as this could affect the correctness of your final answer. E.g. 0.12345/0.23456 = 0.526, while rounded numbers give a different, incorrect answer of 0.123/0.235 = 0.523. Using R : The assignment consists of parts to be solved theoretically and parts to be solved using R . Parts to be solved with R are explicitly identified. Please refer to the R tutorial on LEARN for relevant R material. The R tutorials contains all the necessary commands to complete the assignment. Probability Assignment
QUESTION 1 [34 marks] Maximizing Pawfits Pavlov found the ideal manufacturing plant with your help from the previous assignment. Pavlov recruited his younger brother, Markov 1 , to help him with the manufacturing 2 process. However, Markov needs your help identifying the optimal manufacturing process to maximize profits from the Pavlovian conditioner sales. Each unit of Pavlovian conditioner has its own measure of quality X , which has the probability density function f ( x ) = 2 x λ 2 , 0 x λ. a) [8 marks] The constant λ depends on the manufacturing process, and Markov can alter it. (i) [2 marks] Which values of λ ensure that f ( x ) is a valid probability density function? (ii) [2 marks] Determine E ( X ) , the expected value of the quality of a unit of conditioner. (iii) [4 marks] Set the seed to your student ID. Using R , simulate the quality of N sim = 10 , 000 units of the Pavlovian conditioner when λ = 10 . 3 Print the quality of the first 6 units of conditioner and the mean quality of all 10 , 000 units. b) [8 marks] Markov inspects every unit and any unit for which X < 1 is discarded, while all other products would be sold. (i) [4 marks] Find the probability that a randomly chosen Pavlovian conditioner product is discarded. (ii) [4 marks] By using your simulated data from part a) (iii), calculate the proportion of units that get discarded in the simulation. c) [18 marks] Pavlov’s costs (in dollars) of manufacturing each unit of product are given by c = + b.

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