MAE108S22p5 (1)

MAE 108: Probability and Statistical Methods for Mechanical Engineering Homework 5, Due 23 May 2022 (Monday) to Gradescope by 11:59pm 1. Problem 4.20 Ang and Tang 2. The size of photos on your smart phone is an independent, identically distributed (i.i.d) random variable with a mean of µ X = 3 MB (mega bytes), and a standard deviation of σ X = 0 . 4 MB . You want to store 100 photos on your smart phone. The probability density function of these i.i.d. random variables is the uniform distribution. (a) What is the probability that the sample mean of the 100 photos on your smart phone is between 2.95 and 3.05 MB ? (b) You want to be sure that you have less than a 1% chance of running out of storage on your phone for the 100 photos. What is the size of storage that your phone should have? (c) You decide that you also want to be able to store 40 videos on your smart phone. The videos also are i.i.d. (uniformly distributed) with a mean µ V = 10 MB and a standard deviation σ V = 1 MB . The size of the videos also is independent of the size of the photos. You now want to be sure that you have a less than 1% of running out of storage on your phone for the 100 photos and the 40 videos. What is the new size of storage that your phone should have? 3. Problem 6.1 Ang and Tang For parts c and d, estimate the two sided confidence intervals. 4. Problem 6.3. Ang and Tang 5. Probability density function for wave amplitudes. In the HW module

on Canvas, a data file with 5400 one second samples of wave amplitudes from Ipan, Guam is provided in MAE108S22p5data.mat. (a) Plot a histogram of the variable eta. (b) Estimate the sample mean and sample standard deviation the variable eta. (c) Overlay on your histogram the PDF of a normally distributed random variable using the sample mean and standard deviation found from the data in part (b). The function normpdf is useful for plotting the PDF. 6. Problem 6.7. Maximum likelihood estimator (MLE) for ocean wave heights. Ocean wave heights are a random variable, H that obey the Rayleigh distribution: f H ( h ) = 2 h α 2 exp( − ( h/α ) 2 ) h ≥ 0 (1) f H ( h ) = 0 otherwise where α is a parameter. We showed in lecture that the method of maxi- mum likelihood for h = h 1 , h 2 , ..... h n yields the MLE for α ˆ α 2 = 1 n n X i =1 h 2 i (2) (a) Find ˆ α for the wave height stored in zerouph in MAE108S22p5data (use (2), above). Note that α is called the root mean square (rms) wave height. (b) Plot the normalized histogram of the wave height zerouph. (c) Overlay on the histogram the Rayleigh PDF, (1), using the value of your MLE of ˆ α found in (b). 7. A company is considering putting a new type of coating on a bearing that it produces. From experience, the true average wear life of the bearing with the current coating is µ = 1000 hrs. The company does not want to change production unless evidence strongly suggests that the new coating will increase the true average wear life of the bearing. Prototypes of the bearing with the new type of coating yield wear life data (in hours) of 900 , 1100 , 1225 , 1150 , 1075 ,