HW8_solution

CEE 373 Homework #8 Due date: November 10 at 5pm Total: 100 points 1. (40 points) [Computer required] Before solving this problem, please review the starter code: https://colab.research.google.com/drive/1Qg2waI348FMSqB5cRJv1O2mj9CwpWzHP? usp=sharing Let’s revisit your monthly salary as a newly graduated engineer with your E.I.T. certification in hand from last homework. Again, the starting salary for graduate engineers with their E.I.T. certification is uniformly distributed between $ 70,000 to $ 110,000 per year. You are still interested in X , which is 30% of your monthly salary. Now, you will answer questions about X using Monte Carlo Simulation. (a) Generate 1,000,000 samples of X . Plot these samples in a histogram in comparison with the PDF of X . (b) Consider Y to be the monthly amount your potential roommate can put towards rent. This is independent and identically distributed to X . You are again interested in your combined monthly salary that can go to- wards rent, Z = X + Y . Using Monte Carlo Simulation, generate 1,000,000 samples of Z . Plot a histogram of these samples with the PDF of Z (you already calculated the PDF in HW7). (c) Using your Monte Carlo simulation results, calculate the probability of Z being greater than or equal to $ 4000 ( P ( Z ≥ 4000)). Compare this to the probability you calculated in Homework 7. (d) You believe your sample mean is equal to the true mean of X when you draw multiple samples from X . Employ the Central Limit Theorem (CLT) to cal- culate the true mean ( µ X ) and standard deviation ( σ X ) of the sample mean ( X ) for the following sample sizes: n = { 1 , 2 , 10 , 30 , 300 } . This calculation can be done by hand. (e) For each of the specified sample sizes, n = { 1 , 2 , 10 , 30 , 300 } , perform the following steps. 1. Generate n samples of X 2. Calculate the sample mean X 3. Repeat (1) and (2) k = 1 , 000 times 4. Plot a histogram of your results from (1) - (3) and compare this to the normal PDF using the calculated respective mean and standard deviation from part (e) What do you observe about the comparison between the histogram and the Normal PDF for all values of n ? Page 1 of 11

CEE 373 Homework #8 (f) Do the same as part (e), now using an exponential distribution, E , with λ = 0 . 25 instead of X (g) Now consider that your roommate and your combined income is T = cos 3 ( xy )+ sin ( y ). Using Monte Carlo Simulation, obtain the PDF of T, f T ( t ) and plot. Solution: This is the solution code ( https://colab . research . google . com/drive/1LNI39Lf95pUZTo2M1xLyaIEPHJ sharing ) for the answers and figures. (a) The density histogram of Monte Carlo simulation with 1,000,000 samples and PDF of X. (b) - PDF of Z from the HW 7: f ( z ) = ( 0 . 001 1000 ( z − 3500) (3500 ≤ z ≤ 5500) − 0 . 001 1000 ( z − 5500) (4500 < z ≤ 5500) - Monte Carlo simulation Draw 1,000,000 samples of X and Y from the uniform distribution. X ∼ Uniform (1750 , 2750) , Y ∼ Uniform (1750 , 2750) Z values are the sum of X and Y sampling values. When drawing PDF and density histogram of samples, Page 2 of 11

CEE 373 Homework #8 The shape of density histogram is close to the PDF of Z. (c) - Using PDF: P (4000) = Z 5500 4000 f ( z ) dz = 0 . 875 ( See HW 7 solution ) - Using Monte Carlo simulation: P ( Z ≥ 5000) = the number of samples higher than 4000 the number of total samples ≈ 0 . 875 (The values can be different, but it should be close to 0.875). (d) The mean of PDF: E ( X ) = µ X = Z 2750 1750 xf ( x ) dx = 2250 The variance of PDF: V ar ( X ) = σ 2 X = Z 2750 1750 ( z − µ X ) 2 f ( x ) dx = 288 . 68 2 The mean of sample mean: µ X = µ X = 2250 Page 3 of 11