4.8 Assignment 4

Purposes

The following questions assess your knowledge of the concept of the probability distribution of a discrete random variable, finding the mean (excepted value) and the standard deviation of a discrete random variable as well as applications of binomial distribution.

Instructions

Part A

Complete the following:

1. A variable [latex]X[/latex] of a finite population takes values 5, 7, or 8 and has the following frequency distribution:
   

   [latex]x[/latex]	5	7	8
   frequency	25	40	60

   1. Determine the probability distribution of the random variable [latex]X[/latex]. (3 marks)
   2. Use random variable notation to describe the events that [latex]X[/latex] takes the value 6, a value of at most 6, and a value greater than 6. (3 marks){X = 6}, {X≤6}
   3. Find [latex]P(X = 6)[/latex], [latex]P(X \leq 6)[/latex], and [latex]P(X > 6)[/latex]. (5 marks:1+2+2)
   4. Construct a probability histogram for the random variable [latex]X[/latex]. (3 marks)
   5. Find the mean and the standard deviation of the random variable [latex]X[/latex]. (8 marks: 3+5)
2. An American roulette wheel contains 38 numbers: 18 are red, 18 are black, and 2 are green. When the roulette wheel is spun, the ball is equally likely to land on any of the 38 numbers. Suppose that you bet $1 on red. If the ball lands on a red number, you win $1; otherwise you lose your $1. Let X be the amount you win on your $1 bet.
   1. Determine the probability distribution of the random variable [latex]X[/latex]. (3 marks)
   2. Find the expected value of the random variable [latex]X[/latex]. Interpret the expected value. (5 marks)
   3. Approximately how much would you expect to lose if you bet $1 on red 100 times? 1,000 times? (4 marks)
   4. Is this game fair for the players? Explain. (2 marks)
3. There are four investments. The return on each investment depends on whether next year’s economy is strong or weak (column variable). The following table summarizes the possible payoffs, in dollars, for the four investments (row variable). Let V, W, X, and Y denote the payoffs for the certificate of deposit, office complex, land speculation, and technical school, respectively. Then V, W, X, and Y are random variables. Assume that next year’s economy has a 40% chance of being strong and a 60% chance of being weak.
   

   Determine the expected value of each random variable. (8 marks) Which investment has the best expected payoff? Which is the worst? (4 marks) Which investment would you select? Explain. (3 marks)	Strong Weak Certificate of deposit 6,000 6,000 Office complex 15,000 5,000 Land speculation 33,000 -17,000 Technical school 5,500 10,000

4. A quiz consists of 10 multiple choices questions with five choices A, B, C, D, and E. I did not study and randomly picked one answer for each question.Find the probability that
   1. I get at least one question correct. (4 marks)
   2. I pass the quiz. A passing grade is 60% or better. (5 marks)
   3. I receive an “A” on the quiz (i.e., 90% or better). (3 marks)
   4. How many questions would you expect me to get correct? (2 marks)
   5. Obtain the standard deviation of the number of correct answers. (2 marks)

Part B

Finish the following questions using R and R commander

Use R commander to doublecheck your answers for Question 4 parts (a)–(c). Please copy and paste the computer outputs in the space below. (8 marks: 3+3+2)