PROBLEM 1: The two foremost aerospace companies in the USA are Lockheed and Boeing. They often compete for government contracts. A major factor in awarding a contract is the claimed reliability of the given system to be built. Suppose that a particular space shuttle system health has three levels: (i) good, (ii) bad, but fixable, and (iii) bad.
(a) Construct a 2-D random variable, call it (X, Y), where X relates to the company and Y relates to the system health. Recall that this random variable is not well-defined unless its sample space is also given.
(b) Let A denote the event that the system is bad but fixable. Give this event in (i) standard notation and (ii) as a subset of your sample space in (a).
(c) Let B denote the event that the system was built by Boeing. Give this event in (i) standard notation and (ii) as a subset of your sample space in (a).
(d) Let C denote the event that that Boeing built the system, given that it has bad, but is fixable. Give this event in (i) standard notation and (ii) as a subset of your sample space in (a).
(e) Let D denote the event that that Boeing built the system, and that it has bad, but is fixable. It should be clear that there is a difference between the events C and D. Explain this difference in relation to the concept of a sample space.
(f) Let E denote the event that that the system is not good, given that it was built by Lockheed. Give this event in (i) standard notation and (ii) as a subset of the associated constrained sample space.
PROBLEM 2: In an effort to quantify the likelihood that a given manufactured part will or will not meet a given specification, a sample of n=10 parts will be randomly selected and measured for conformance. The guiding random variable, X, is the act of noting whether or not any selected part conforms to the specification, with the events [X = 0] ∼ conforms and [X = 1] ∼ does not conforms. Let Pr[X = 1] =Δ Px. Let X = (X1, X2, · · ·, X10) denote the 10-D data collection variable associated with X.
(a) How many elements are in the sample space SX?
(b) Let Y = k-1∑10Xk denote the act of recording the number of sampled parts that do not conform. Give SY.
Remark. Because the elements of X = (X1, X2, · · ·, X10) are data collection variables in relation to X, each element is identically distributed (id), having the same distribution as X. Furthermore, because the elements are selected randomly, it is reasonable to assume that they are mutually independent (i). In other words, the elements of X are iid random variables. It should be clear that k-1∑10Xk =Δ Y∼ binomial (n=10, p) and
.
(c) Even though the goal of the study is to estimate px, assume in this part that, in truth, Px = 0.1. Use the Matlab command 'binomd' to run nsim = 15 simulations of P^x = Y/10 = X-. Include your code in the Appendix @ 2(c), and copy/paste your results below. [Give each number to only ONE decimal place.]
(d) Even though, in truth, px = 0.1, you should have found in (c) that some of your 15 simulations yielded estimates p^x ≠ px. This should be expected, since the estimator p^x is, after all, a random variable. Use the command 'binopdf' to compute Pr[p^x = 0]. Copy/paste the command and result HERE.
(e) In view of your answer in (d), explain why you think (or do not think) your simulation results in (c) are reasonable.
(f) Use the command 'binopdf' to search for the value of the smallest sample size, n, that would yield Pr[p^x = 0] ≤ 0.05. Give the value of the associated probability, as well as the probability for sample size n-1.
(g) To verify your answer in (f), use the pdf expression
to solve for the numerical value of n (it will not be an integer), such that Pr[p^x = 0] ≤ 0.05.
PROBLEM 3: Suppose that a given system has n=10 critical components. Let (X1, X2, · · ·, X10) denote the states of these components, where. Let [Xk = 0] and [Xk = 1] denote the events that the kth component is good and bad, respectively. Let k-1∑10Xk denote the number of bad components.
(a) Assume, as is the case in most textbooks on the subject, that these ten random variables are mutually independent and identically distributed (iid), and for each k ∈{1, 2, · · · , 10} we have pk =Δ[Xk = 1] = p. It should be clear that, in this case, Y ∼ binomial (n = 10, p). For p=0.05 use the Matlab command 'binopdf' to ultimately arrive at a stem plot of fY(y) over the range y ∈ {0, 1, · · · , 10}.
(b) Use your plot in (a) to visually estimate Pr[Y ≤ 2].
(c) Use the appropriate Matlab command to obtain the value of Pr[Y > 4]. Include your command HERE.
(d) The above parts related to TRUE probabilities. However, often those probabilities are not known, and must be estimated from data. Use the Matlab command 'binornd' to simulate measurements of the data collection variables {Yk}k-15. (i) Include your command and the resulting 5 measurements {yk}k-15 HERE.
(e) (i) Use your data in (d) to estimate Pr[Y = 0]. Then (ii) discuss how your estimate compares to the true value of Pr[Y = 0].
(f) Denote your estimator in (e) as Pr^[Y = 0]. To better understand this estimator, begin by using the 'binornd' command to simulate 104 estimates of Pr^[Y = 0]. Then use the Matlab commands 'mean' and 'std' to arrive at simulation-based approximate values for the mean and standard deviation of Pr^[Y = 0]. Give these numbers here. Include your code @ 2(f) in the Appendix.
PROBLEM 4: There are many endeavors in life that must be repeated again and again, until you 'get it right'. Let [Xk = 0] and [Xk = 1] denote the events that you get it wrong, or right, respectively, on the kth attempt, and let Pr[Xk=1]=Δ pk. Let the event [Y = y] denote the event that you finally get it right on the yth attempt.
(a) Assume that the collection of random variables {Xk}k-1∞ are mutually independent and identically distributed (iid) random variables, and that pk = 0.1 (i.e. the endeavor is a relatively challenging one). Under these assumptions, it should be clear that Y ∼ geometrixc. Use the Matlab command 'geopdf' to arrive at a stem plot of fY(y) for the range of attempts y ∈ {1, 2, · · · , 50}. Include your code @ 4(a) in the Appendix.
(b) Use the Matlab command 'geocdf' to arrive at the value for Pr[Y > 5].
(c) Identify the equations in the book that give the mean and standard deviation for Y, and compute their numerical values.
(d) The gain some experience in simulating random variables, use the Matlab command 'geornd' to simulate a single measurement of the data collection variables {Yk}k-110. (i) Copy/paste the command and related 10 measurements {yk}k-110 HERE.
(e) Now, still assume that {Xk}k-1∞ are mutually independent, but that in the beginning stages, after any given failure you increase your determination such that the probability of success on the next attempt is 50% greater than that of the last attempt. Assume that p1 = 0.1. Arrive at Pr[Y > 5]. Show all equation development HERE, but give related code @ 4(e) in the Appendix.