I want to use R markdown to do the following questions and render a pdf for all the answers!!!

Q1.

Suppose we toss 4 coins (each having heads probability = (1/2). Let X denote the random variable: (number of heads) - (number of tails).

(a) What is the range of X? (give exact upper and lower bounds along with a line of explanation)

(b) What is the probability mass function of

(c) What is the cumulative density function of X

Q2. Using R, simulate tossing 4 coins as above, and compute the random variable X defined in problem 1. Estimate the probability mass function you computed in part (b) of problem 1 by simulating 1000 times and averaging.

At a Midwestern business school, historical data indicates that 70% of admitted MBA students ultimately join the business school’s MBA program. In a certain year, the MBA program at the University admitted 200 students.

a. Find the probability that at least 150 students ultimately join the MBA program.

b. Find the probability that no less than 135 and no more than 160 students finally join the MBA program.

c. How many students should the MBA program expect to join the program?

d. What is the standard deviation of the number of students who will join the MBA program? e. Let X be the number of students out of 200 who will join the program. Would the empirical rule apply to the probability distribution of X in this case?

The voting paradox (10 points) Consider an election with three candidates, George, Lee, and Ray. There are 12 voters with the following preferences:

In the plurality voting, where 12 voters vote for the three candidates and each voter is allowed to vote for only one candidate, who will become the winner? How many voters actually prefer G over R and how many voters prefer R over G? What are the advantages and disadvantages of the plurality voting?

2). In the pairwise voting following the majority rule, is there any candidate who is a dominate winner (one wins in every pairwise comparison) or a sure loser (one loses in every pairwise comparison)? Explain the voting paradox in this example if there is any.

Using the below program as a starting point

C++ Program - Arrays- Chapter 9

    Include the following header files in your program:     string, iomanip, iostream
Suggestion: code steps 1 thru 4 then test then add requirement 5, then test, then add 6, then test etc. Add comments to display assignment //step 1., //step 2. etc. This program is to have no programer created functions. Just do everything in main and make sure you comment each step so I can grade more easily.
C++ Program Pointers
Before starting on this project you should make sure points 1 to 15 are working properly from Chapter 7 Arrays project.
This program is to have no programer created functions. Just do everything in main and make sure you comment each step so I can grade more easily.
C++ Program Extension - This will be graded as a separate program but will add the following to Chapter 7 Arrays project. Don't forget to include comments with the corresponding number.

Extend the Array project to include:
16. Define a pointer to a double, pdArray.
17. Assign the pointer, pdArray, to contain the address of the double array, dArr:
18. Use the array name, dArr, to print out the array elements with subscript notation, [ ]. All on 1 line a space between each.
19. Use the pointer to print out the array elements with pointer notation while not changing the pointer itself. Use a for loop. *( pdArray + Cnt1) would be an example. All on 1 line a space between each.
20. Use the pointer to print out the array elements with pointer notation but change the pointer to point to the actual array element rather than the method in 18. All on 1 line.
*pdArray would do this if the loop has the following post loop operation:   pdArray++
21. Use the array name for the double array and pointer notation to print the entire array, all on one line.
22. Using a different pointer, piArray, allocate enough memory for 100 int's and assign the address to the pointer.
23. In a for loop assign every item in the array to be a random number from 1 to 49 ( hint: rand() % 6 + 1 gives random numbers from 1 to 6 )
24. Using cout print the first 10 items in the array, all on 1 line.

USE THIS PROGRAM AS STARTING POINT

//Christopher Cupani
//Sept 22, 2019
//Online class
//Chapter 7
#include <iostream>
#include <istream>
#include <string>
#include <iomanip>
#include <cstring>
using namespace std;

int main()
{
   //The variables are declare
   double dArr[5];
   double lArr[7] = { 100000, 134567, 123456, 9, -234567, -1, 123489 };
   int iArr[3][5];
   char sName[30] = { 'C','h','r','i','s' };
   //define variables cnt1 and cnt2 (short data types)
   short cnt1, cnt2;
   long double total = 0;
   //define one long variable and it call highest
   long highest;
   // 4
   int i;
   //Create a loop to put a random number into each of the elements
   //of the array of double.
   srand(time(0));
   for (i = 0; i < 5; i++) {
       double f = (double)rand() / RAND_MAX;
       dArr[i] = 1 + f * (49);
   }
   //Create a for loop to display all of the values in dArr.
   for (i = 0; i < 5; i++) {
       cout << dArr[i] << " ";
   }
   cout << endl;
   // loop to add up the array of double, dArr, into the
   //variable total
   for (i = 0; i < 5; i++) {
       total += dArr[i];
   }
   //display the total
   cout << "Total of double array is " << total << endl;
   //display the average
   cout << "Average of double array is " << total / 5 << endl;
   // Create a for loop that was like example in the instructions
   //to the following for the long array, lArr.
   for (cnt1 = 1, highest = lArr[0]; cnt1 < 7; cnt1++)
   {
       //logic to compare each array element, starting with lArr[1], with highest
       //replace highest if the value in lArr[cnt] is higher than the value in variable highest
       if (highest < lArr[cnt1])
           highest = lArr[cnt1];
   }
   //display the highes value
   cout << "Highest is " << highest << endl;
   //generate random number in 2d array between 1 to 53
   for (cnt1 = 0; cnt1 < 3; cnt1++)
   {
       for (cnt2 = 0; cnt2 < 5; cnt2++) {
           iArr[cnt1][cnt2] = rand() % (53) + 1;
       }
   }
   cout << endl;
   //Display the 2 d array with setw(3)
   cout << "iArr is " << endl;
   for (cnt1 = 0; cnt1 < 3; cnt1++)
   {
       for (cnt2 = 0; cnt2 < 5; cnt2++) {
           cout << setw(3) << iArr[cnt1][cnt2];
       }
       cout << endl;
   }
   cout << endl;
   //Display 2d array iArry column wise
   cout << "iArry print in column wise " << endl;
   for (cnt1 = 0; cnt1 < 5; cnt1++)
   {
       for (cnt2 = 0; cnt2 < 3; cnt2++) {
           cout << setw(3) << iArr[cnt2][cnt1];
       }
       cout << endl;
   }
   cout << endl;
   //Use cin.getline( ...... ) to add another name into variable sName.
   cout << "Enter the name " << endl;
   cin.getline(sName, 30);
   //Display the name
   cout << sName << endl;
   cnt1 = 0;
   //Display the ascii value of each character in the char array,
   //1 per line. Use a while loop to look for the '\0' as a signal to end.
   while (sName[cnt1] != '\0') {
       cout << sName[cnt1] << " Ascci is " << int(sName[cnt1]) << endl;
       cnt1++;
   }
   //Entering Albert Einstein to sName array and
   //using strcpy_s function.
   strcpy_s(sName, "Albert Einstein");
   cout << sName << endl;
   //Display the ascii of 12th character
   cout << "12th character ascii value is " << int(sName[12]);

   return 0;
}

An SAT test has 8 questions, each question having 5 multiple choice answers. If a student make random guesses for each answer, find the probability that: a)the number of correct answers is 7. b)the number of correct answers is at least 4. c)the number of correct answers is fewer than 3.

Based on a poll, among adults who regret getting tattoos, 13% say that they were too young when they got their tattoos. Assume that ten adults who regret getting tattoos are randomly selected, and find the indicated probability.

A. Find the probability that none of the selected adults say they were too young to get tattoos. (Round to four decimal places as needed).

b. Find the probability that exactly one of the selected adults says that he or she was too young to get tattoos. (Round to four decimal places as needed.)

c. Find the probability that the number of selected adults saying they were too young is 0 or 1. (Round to four decimal places as needed.)

d. If we randomly select ten adults, is 1 a significantly low number who say that they were too young to get tattoos?

(Yes, No) because the probability that (at most 1, less than 1, at least 1, more than 1, exactly 1) of the selected adults say that they were too young is (greater than, equal to, less than) 0.05

Global TV implements Varian’s Position Auction for commercial spots during the evening news. The highest bidder is to be awarded the first commercial spot which is known to attract 2000 viewers and the second-highest bidder is to be awarded the second commercial spot which is known to attract only 500 viewers. They use a position auction (GSP) mechanism: the highest bidder will pay a price equal to bid of the second-highest bidder and the second-highest bidder will pay a reserve price of 2 dollars. There are two bidders: Dodge Motor Company who have a valuation of 6 dollars per viewer and Ford Motor Company who have a valuation of 4 dollars per viewer. what is the outcome of this bid?

The number of months that a light bulb is functioning is exponentially distributed with parameter λ = 1/10 .

What is the probability that a light bulb will burn for at least 4 months?

Find the mean and the standard deviation.

What is the probability that a light bulb will burn for at least 12 months, given that it has burned for 4 months?

In a family with 6 children, excluding multiple births, what is the probability of having exactly 2 girls?
Assume that having a boy is as likely as having a girl at each birth.

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A biotechnology company produced 201 doses of somatropin, including 9 which were defective. Quality control test 15 samples at random, and rejects the batch if any of the random samples are found defective. What is the probability that the batch gets rejected?

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A math professor finds that when she schedules an office hour for student help, an average of 3.2 students arrive. Find the probability that in a randomly selected office hour, the number of student arrivals is 5.

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The mean number of patients admitted per day to the emergency room of a small hospital is 2. If, on any given day, there are only 9 beds available for new patients, what is the probability that the hospital will not have enough beds to accommodate its newly admitted patients?

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In a family with 6 children, excluding multiple births, what is the probability of having exactly 2 girls?
Assume that having a boy is as likely as having a girl at each birth.

---------------------------------

A biotechnology company produced 201 doses of somatropin, including 9 which were defective. Quality control test 15 samples at random, and rejects the batch if any of the random samples are found defective. What is the probability that the batch gets rejected?

--------------------------------

A math professor finds that when she schedules an office hour for student help, an average of 3.2 students arrive. Find the probability that in a randomly selected office hour, the number of student arrivals is 5.

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The mean number of patients admitted per day to the emergency room of a small hospital is 2. If, on any given day, there are only 9 beds available for new patients, what is the probability that the hospital will not have enough beds to accommodate its newly admitted patients?

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