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A three-sided die has three faces, one face stamped with one dot, a second face stamped with two dots, and a third face stamped with three dots. When the die is rolled, one of the three faces is displayed. In many trials of rolling the die, it is found that the face having one dot is displayed 10% of the time; the face having two dots is displayed 30% of the time, and the face having three dots is displayed 60% of the time.
A random experiment consists of rolling two identical three-sided dice; the experimental outcome is recorded as a two-tuple (-,-) where the first entry identifies the number of dots showing on die 1 and the second entry identifies the number of dots showing on die 2. Answer the following questions concerning this random experiment.
A. We first investigate a probability space P = (S, E, P) for this experiment.
1. Identify the sample space S.
2. Identify the number of events comprising the sigma algebra of events E. Do not attempt to list all events in the event space E.
3. Identify the value that the probability measure P should assign to those events in E corresponding to the experiment outcomes. Do not attempt to list the value that P assigns to each event in the entire event space E.
B. A random variable X is defined by assigning a real number to each experimental outcome. The real number is ***** to the sum of the number of dots stamped on each of the two displayed die faces.
1. Write an expression for and accurately draw and label the cumulative probability distribution function F(x) for the random variable X.
2. Write an expression for and accurately draw and label the probability density function p(x) for the random variable X.
Problem Two
use an Excel spreadsheet to generate an independent set of 100 random numbers. Use the “2D Column Plot” feature to develop a five-interval approximation to the probability density function for this random variable. Submit your Excel spreadsheet with your solution.
Problem Three
In this problem, we investigate the generation of samples from a random variate in a spreadsheet environment. Let X be a random variable having exponential probability density function p(x) where
.
For expected mean , we wish to generate 100 samples from X using an Excel spreadsheet.
In the latest version of Excel, there is a statistical formula that can be used to generate these samples. However, in earlier versions, it was necessary to utilize the inverse function theorem to generate these samples. In this problem, you are to utilize the inverse function theorem.
A. Describe the process that you will use to generate independent random samples from the random variable X. Derive any formulas that you will need to implement in Excel.
B. Implement the process that you describe in Part A on an Excel spreadsheet and then generate 100 sample points from X. Submit your Excel spreadsheet with your solution.
C. Use the “2D Column Plot” feature to develop a five-interval approximation to the probability density function for this random variable. Paste your Excel plot into your solution document. In addition, submit your Excel plot and associated computations on the same spreadsheet used for Part B.