ENGM2032, Applied Probability and Statistics Fall 2016 Assignment #6 Due: Thu Nov 3, 2016 1) Suppose that 30% of stocks (from different companies) purchased by a certain stockbroker have first year returns that exceed expectations. a) What is the approximate probability that, out of the next 100 stocks purchased by the broker, more than 40 will have first year returns exceeding expectations? b) How many stocks should the broker purchase to be 95% sure of having at least 30 with returns exceeding expectations? c) If the broker has purchased 100 stocks, then the broker will be 90% confident of having at least how many of these stocks with returns exceeding expectations? 2) Suppose that the structural load applied to a column at each story of a structure is uniformly distributed between 24 and 36 kN. If the structure is 30 stories tall and loads are assumed to act independently, then a) what is the probability that the total load at the base of the column exceeds 950 kN? b) if the foundation at the foot of the column is to be designed using a load having only a 5% chance of being exceeded in this structure, what load should be used in the design? 3) A concrete manufacturer wants to implement \"quality control\" of the concrete mixture so that only 5% of the concrete has compressive strength below 30 MPa (for safety) and only 10% of the concrete has compressive strength above 36 MPa (for cost efficiency). Assuming compressive strength is normally distributed, fX (x) = p1 2 e (x)2 2 2 then what values of and will provide such control? Note that in practice, is set by the ingredients used in the mixture (the design), while is determined by the amount of care taken in selecting, measuring, and mixing the ingredients (quality control). That is, quality control is usually aimed at reducing . 4) Effluent from a chemical plant is passed through a clearing component that actually consists of two filters. Once the first filter becomes clogged with sediment, the flow is diverted through the second, nominally identical, filter. Once the second filter becomes clogged, the entire component is replaced. Suppose that the number of cubic metres of fluid cleared by a single filter before it becomes clogged is exponentially distributed with mean 10,000 m3 and that filters become clogged independently. a) What is the probability that a filter will clean less than 8,000 m3 of fluid before becoming clogged? b) What is the probability that a clearing component will need to be replaced prior to its clearing 16,000 m3 of fluid? c) If we start measuring the fluid volume from the time that the first clearing component is put in place, what is the probability that the first, and only, clearing component is replaced prior to clearing 16,000 m3 of fluid? ENGM2032, Applied Probability and Statistics Assignment #6 Fall 2016: pg 2 5) The bearing capacity of a soil to support a load depends on the weakest shear failure path through the soil. When the soil fails, it squeezes out from under the load by sliding along this weakest shear failure path. Suppose that tests have shown that the bearing capacity of a particular soil type follows a Weibull distribution with = 2 and that 90% of the soils have bearing capacities in excess of 2 MPa. What is this soil's median bearing capacity? 6) The head of an engine is held in place by 6 bolts. The engine will fail if any of the bolts fail - that is, there is no redundancy in the system. Suppose that each bolt has an exponentially distributed lifetime with mean 5 years and assume that bolts fail independently (even though this is a questionable assumption). Let Ai be the event that the ith bolt lasts at least t years and let X be the random variable representing the time at which the engine fails due to head bolt failure. a) What is the event fX > tg equivalent to, in terms of the Ai 's? \u0002 b) Using the independence of the Ai 's, compute P X for X. What distribution does X have? \u0003 \u0014 t . Then obtain the pdf and the cdf 7) Suppose that the tensile strength of a particular wood is lognormally distributed with mean 10 MPa and coefficient of variation 25%. If a random sample of this wood is tested, a) What is the probability that its tensile strength will lie between 6 and 12 MPa? b) Suppose that the design tensile strength of this wood is its 10th percentile. What is this wood's design tensile strength? (Note: the 10th percentile is the value, x10 , such that \u0002 \u0003 P X < x10 = 0.1)