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Part 3. (13 polnts) SImulatlon of Gamma Random Varlables Background: When we use the probablity denslty function to find probabiltles for a random varlable, we are using the density function as a model. This Is a smooth curve, based on the shape of observed outcomes for the random varlable. The observed distribution will be rough and may not follow the model exactly. The probabillty density curve, or function, is still just a model for what is actually happening with the random variable. In other words, there can be some discrepancies between the actual proportion of values above x and the proportion of area under the curve above the same value x. Our expectation is as the number of observations increase, literally or theoretically, the observed distribution will align more with the density curve. Over the long run, the differences are negligible, the model is sufficlent and more convenient to find desired information Simulation: Use R to simulate 1000 observations from a gamma distribution. To begin alpha 2 and beta = 7. Highlight and run the parameters and observation values. Run the simulation code to plot the observations and fit the probability density function over the observations. You don't need to change anything. You may run the section all at once by highlighting all of the section and running it by clicking the run button at the top of the script window. a. Given the values are from a gamma distribution with alpha- 2 and beta -7, i. li. (1 points) What is the expression for the probability density function? (1 point) What is the average and standard deviation of random variable? Show work. li. (1 point What is the probability x is less than 47 Show work. b. (2 point) Run the simulation and paste your plot. Comment on the general shape of the distribution. How well does the density curve fit the observations? (2 point) What is the exact proportion of values below 4? How does the actual proportion compare to the probability from the density curve in part 2-a-ii? (1.5 point) Increase the number of observations to 10000, rerun the simulation. Paste your plot. How does increasing the number of observations affect the fit of the density curve? (1.5 point) What is the exact proportion of values below 4? How does increasing the number of observations affect the accuracy of the model? Make a comparison between this proportion and 2-a-ili and 2c. c. d. e. f. (1 point) Rerun the simulation with alpha 1, beta 7, and observations 10000. Paste your plot. Comment on the general shape of the distribution. (2 points) This model is a special case of the gamma distribution, what is it specifically? What is the expression for the probability density function? g.

Answer 1