Part I: Random Variables and Probability Distributions 1. Let X equal the sum of dots showing when you roll two dice. The variable X can take on 11 possible values from 2 to 12, with the following probabilities: Relative Xj Pi Pixi | Frequency, ni Frequency, fi fixi 1/36 3 2/36 4 3/36 4/36 6 5/36 6/36 8 5/36 9 4/36 10 3/36 11 2/36 11 12 1/36 Total: 36/36 100 100/100a. Calculate u, the mean {or expected value) of X, and 0', the standard deviation of X. b. Now in R, we simulate throwing a pair of dice 50 times (n=50). Please run the script \"code_assignment_4_diee.R\". Then, in the column headed Frequency (j) record the number of times the sum of the dots showing on the two dice sums to 2, 3, etc. c. Calculate the Relative Frequency of X {fj = min) for these 50 observations of X, ll in the table. The R script also calculates this, and you can use that to check your answers. d. Plot the distribution of your simulation of 50 throws. e. Calculate i , the mean of X, for your 50 throws. You can think of this as an estimate of u, the true mean of X, based on a sample of 50 observations. Is your estimate of the mean of X reasonably close to the expected value of X? Please report i and also report down two others values: A= X + 0.67, and B= i - 0.67. {W e will be using the values of A and B in a lture lab). Does your range A to B include the population mean