Discrete and continuous distributions.

    The Suez Canal is a vertical straight thoroughfare broken up into three segments: Begins at, degrees latitude Ends at, degrees latitude The Red Sea stretch 29.90 30.17 The Great Bitter Lake stretch 30.17 30.80 The Mediterranean stretch 30.80 31.26 The Great Bitter Lake stretch is two-way but the other two stretches are one-way. So, if you make it into a one-way stretch, you’ll cruise through the canal safely. But if you’re caught in the middle stretch when the authorities reverse the traffic in the one-way stretches, you’ll be stranded in the middle of the canal waiting till the traffic is your way again. Ships are distributed uniformly through the canal. (a) Find the probability density of the canal, report its units. (b) Draw the graphs of the PDF, CDF, and quantile functions for this case. Label the graphs, the axes, and the values you find relevant (c) What is the probability of being trapped in the middle segment? Find the CDF of the gate between The Red Sea and The Great Bitter Lake. Then find the CDF of the gate between the Great Bitter Lake and the Mediterranean. Then use those values to find your probability. 2. Normal distribution Now that car companies seal transmissions, they either run or require complete replacement, no intermediate solutions. Because of this, the expected life of a transmission becomes a matter of life and death. On average, a transmission lasts 15 years with a standard deviation of two years. The lifespans are distributed normally. (a) Your car’s transmission lasted 13 years. Draw the PDF, CDF, and the quantile distributions of the situation and your car in it. Label all graphs, values, and axes (b) What is the probability that your next car’s transmission will last between 11 and 15 years? Find the larger and the smaller CDF and find your probability as the difference. Illustrate your work with a PDF graph, on which clearly identify the probability you found. (c) What is the probability your next car’s transmission will last longer than 17 years? Illustrate your answer on the PDF. (d) Draw the story of your 13 year old transmission in terms of the standard normal distribution. Label all relevant values and axes.