STM2EPS : R studio - R Programming Assignment

Question :  1.Suppose that a company produces ball bearings. Each ball bearing is quality tested on several variables such as starting and running torque. If a ball bearing does not pass certain benchmarks on the testing variables then it is deemed to be defective. Let D denote the event that a ball bearing is truly defective and let F denote the event that a ball bearing fails the quality test. It is known that 1.5% of all ball bearings are truly defective and the probability that a defective ball bearing fails the quality test is 0.95. Unfortunately there is also a 5% chance that a non-defective ball bearing fails the quality test. (a) What is P(D) and P(Dc)? (You are being asked for a numerical answer here, not a worded description of what these represent. If we require a worded description, as we do in the next question, this will be made clear). (b) In words, explain what P(F|Dc ) denotes? (I.e. an answer along the lines of ‘This is the probability that ...’). (c) What is P(F|D) and P(F|Dc )? (d) What is the probability that a ball bearing will fail the quality test? You will need to use the Law of Total Probability to answer this question. (e) Given that a ball bearing fails the quality test, what is the probability that it is truly defective? 2. Suppose that a system of components features two subsystems. Subsystem 1 consists of three identical compo-nents in parallel. Subsystem 2 consists of two identical components arranged in parallel. These two subsystems are then arranged in series. Let R1 denote the reliability of the component used in Subsystem 1 and R2 denote the reliability of the component used in Subsystem (a) Write down the reliability function for Subsystem 1 in terms of R1. (b) Write down the reliability function for Subsystem 2 in terms of R2. (c) Write down the reliability function for the entire system consisting of the two subsystems in series. Call this reliability function R. 3. Recall from Lab Week 3, that in considering ship detection using Product of Multilook Amplitudes (PMA) detectors, Gao et al. (2016) define a probability density function that describes the PMA variable denoted X. For simplicity, for this question let ? = 1 and n = 2 so that this density is..... 4. Let X be the number of cars detected by a pressure plate starting to cross a small bridge in a 10 second interval. Let Y denote the actual number of cars starting to cross the bridge in the 10 second interval. The joint probability mass function for (X, Y ) is below.... (a) What is P(X = 1 and Y = 1) and what does this probability represent? (b) Derive the marginal probability mass function for the random variable Y . (c) Given that two cars start crossing the bridge, what is the probability that the pressure plate correctly detects two cars? (d) Given that no cars start crossing bridge, what is the probability that the pressure plate detects one or two cars?