Quan2600. Statistics I.

HW 7. Discrete and continuous distributions.

Due on 03/21/2022

1. Uniform distribution

The Suez Canal is a vertical straight thoroughfare broken up into three segments:

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.

3. Approximating a binomial with a normal distribution

You go to see eight movies every year and win whenever nobody talks. The long-term probability of nobody talking is , the number of trials in an experiment is .

(a) Find the mean, the variance, and the standard deviation of this binomial distribution

(b) State the values of the smallest and the largest values of that are possible in this distribution.

(c) Draw an approximate density function for a normal distribution described by the parameters you found in (a). Add the values you indicated in (b) to the sketch.

(d) For the normal distribution, described by the parameters you found in (a), find the values of the z scores of -3 and +3.

(e) Use your results from (b), combined with the method from 34:20 of the Ch6 part 2 video, to decide if it is a good idea to approximate this binomial distribution with a normal.

(f) Excel: follow the example at 18:00 in Ch6 part 2 lecture video to put the binomial distribution bars and the normal distribution bell curve on one same Combo chart. Use the values established in (a) – (e). Label axes, the graph itself, and all you deem relevant. Print out the graph to a nice .pdf or attach neatly to your main submission file.

4. Exponential distribution

Boeing 737 is one of the most prolific, oldest, and most polished airliners in the world. In the last 20 years, an airplane of this type has been experiencing 1.2 incidents with more than 50 casualties per year.

Two years ago, the world was bewildered by the two back-to-back crashes of the latest Boeing 737 Max 8 that, together, resulted in more than 300 casualties. Lion Air Flight 610 crashed in October 2018. Ethiopian Airlines Flight 302 crashed in March 2019.

What was the probability that less than five months would pass between the two crashes?

(a) Establish the units in which the length of interval, , is measured in this example. Then, express the long-term average in the same units.

(b) Use the cumulative distribution function formula for the exponential distribution to find the probability that x is less or equal to 5.

(c) Draw the probability density function for the problem, point out , , and on it. Indicate what is on the axes.

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