Statistics_Project_Final__1_.docx MAT-154 Statistics Project Final Problem: Machines

Statistics_Project_Final__1_.docx    MAT-154  Statistics Project Final  Problem: Machines have the probability of falling 1/6 every year. I need to simulate how long will it take to fail in average of 20 times.  Part I (Empirical Analysis):I simulated this by rolling the dice until I obtained 1 every time. The average of the numbers I got is [4.7].This means that in average, it will take about 4.7 years until the machine fails. The numbers I obtain in each round until I got a 1 are [1,3,6,1,2,1,4,9,5,14,21,2,5,3,3,3,2,2,2,5]. I took the difference of these values from the mean [-3.7, -1.7, 1.3, -3.7, -2.7, -3.7, -0.7, 4.3, 0.3, 9.3, 16.3, -2.7, 0.3, -1.7, -1.7, -1.7, -2.7, -2.7, -2.7, 0.3]. Each of these values squared are [13.69, 2.89, 1.69, 13.69, 7.29, 13.69, 0.49, 18.49, 0.09, 86.49, 265.69, 7.29, 0.09, 2.89, 2.89, 2.89, 7.29, 7.29, 7.29, 0.09]. The sum of all of the squares are [462.2], and the divided by the sample size (20) gave me [23.11]. Taking the square root of this give me [4.807] which is the standard deviation. In conclusion, it takes 4.7 years on average before the machine fails, and the low standard deviation tell us that it is reliable because the difference between numbers is small.   Part II (Theoretical Analysis-Approximations):  Now, we havethe calculations of 20 rounds. With this information we are able to calculate an approximation of an infinite number of simulations. This includes Expected value, Variance, and Standard deviation of the variable X(Likelihood of machine to fail). The pr