Write a program that simulates the rolling of two dice. The program should use rand to roll the first dice and should use rand again to roll the second dice. The sum of the two values should then be calculated. Note: Since each dice can show an integer value from 1 to 6, then the sum of the to values will vary from 2 to 12, with 7 being the most frequent sum and 2 and 12 being the least frequent sums. Below shows the 36 possible combinations of the two dice. Your program should roll the two dice 36,000 times. Use a single-subscripted array to tally the numbers of times each possible sum appears. Print the results in a tabular format. Also, determine if the totals reasonable (i.e. there are six ways to roll a 7, so approximately one sixth of all the rolls should be 7).

Exercise III: Catalan numbers For n e N, denote by c, to be the number of ways to form a "mountain range" with n upstrokes (U) and n downstrokes (D) that all stay above a horizontal line. For instance: for n = 1, only UD is allowed, so c = 1; • for n = 2, only UUDD and UDUD are allowed, so c2 = 2. 1. Check that c3 = 5 by writing down or drawing all possible options. 2. Consider the power series +00 g(x) = 2 n=0 (by definition co = 1) called the generating function of the sequence (en). Justify that cn < 4" for each n, and deduce that the radius of convergence of g is at least 1/4. 3. It can be show that for r E (-1/4, 1/4), g(x) = 1+ xg(r)? and therefore 1- VI 4.x g(x) = 2.x Use this formula and the known power series of V1+ x to write the first terms of the power series expansion of g, in the form g(x) = co + c1x + c2x2 + C3x + c4x* + ·.. Show how you obtain a few terms, but you do not need to show all computations and you can use a calculator for fractions. Check that you recover co,…

In various applications, you are often asked to compute the mean and standard deviation of data. The mean is simply the average of the numbers. The standard deviation is a statistic that tells you how tightly all the various data are clustered around the mean in a set of data. For example, what is the average age of the students in a class? How close are the ages? If all the students are the same age, the deviation is 0. Write a program that prompts the user to enter any number of values into a double array, and then calculates and displays the mean and standard deviations of these numbers using the following formulas: mean)? 2i +x2 + .. + xn i=1 i=1 mean = deviation = п - 1

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