INDU 6331 Assignment 2 Advanced Quality Control

A mechatronic assembly is subjected to a final functional test. Suppose that defects occur at random in these assemblies, and that defects occur according to a Poisson distribution with parameter λ= 0.02 a. What is the probability that an assembly will have exactly one defect? λ=0.02 p ( x )= e −λ λ x x ! p( x=1)= e −0.02 0.021 1! =0.0196=1.96 There is a probability of 1.96% that an assembly will have exactly one defect b. What is the probability that an assembly will have one or more defects? p(x=1) = 1 – p(x=0) p( x≥1)=1− e −0.020.020 0! =1−0.98019=0.0198 The probability that an assembly will have one or more defects is 1.98% c. Suppose that you improve the process so that the occurrence rate of defects is cut in half to λ= 0.01. What effect that this has on the probability that an assembly will have one or more defects? P(x=1) = 1 – p(x=0) p( x≥1)=1− e −0.010.010 0! =1−0.99=0.00995 The probability that an assembly will have one or more defects considering λ= 0.01, is 0.995%. Cutting in half the rate of defects reduces the probability calculated to its half. 3.26 The probability distribution of x is f(x)= ke-x , 0≤ x ≤∞. Find the appropriate value of k. Find the mean and variance of x. This study source was downloaded by from CourseH on :11:11 GMT -05:00 INDU 6331 Assignment 2 Advanced Quality Control For this function the appropriate value of k is k= 1. With this value f(x) will describe an exponential distribution: f ( x )=k e −kx , k=1 The mean of this distribution is μ= 1 k = 1 1 =1 And the variance is σ 2= 1 k 2= 1 1 =1 3.33 A random sample of 50 units is drawn from a production process every half hour. The fraction of nonconforming product manufactured is 0.02. What is the probability that Ṗ≤ 0.04 if the fraction nonconforming really is 0.02? Binomial distribution n= 50 p= 0.02 Ṗ≤0.04 Ṗ= x n ≤0.04→ x 50 ≤ 0.04→x ≤2 Probability of finding 2 or less nonconformings in the sample