Stat 2000, Section 001, Homework Assignment 10 (Due 11/13/2002 11:59pm)

• 0) Reading: Sections 4.3, 4.4
  
  
• 1) Please work on the following textbook exercises from Moore/McCabe (3rd Edition):
  
  
  • Exercise 4.40 (2 Points):
    Choose an American household at random and let the random variable X be the number of persons living in the household. If we ignore the few households with more than seven inhabitants, the probability distribution of X is as follows:

    Inhabitants	1	2	3	4	5	6	7
    Probability	0.25	0.32	0.17	0.15	0.07	0.03	0.01

    (a) Verify that this is a legitimate discrete probability distribution and draw a probability histogram to display it.
    (b) What is P(X >= 5)?
    (c) What is P(X > 5)?
    (d) What is P(2 < X <= 4)?
    (e) What is P(X not equal to 1)?
    (f) Write the event that a randomly chosen household contains more than two persons in terms of the random variable X. What is the probability of this event?
    
    
  • Exercise 4.46 (2 Points):
    Many random number generators allow users to specify the range of the random numbers to be produced. Suppose that you specify that the range is to be 0 <= Y <= 2. Then the density curve of the outcomes has constant height between 0 and 2, and height 0 elsewhere.
    (a) What is the height of the density curve between 0 and 2? Draw a graph of the density curve.
    (b) Use your graph from (a) and the fact that probability is area under the curve to find P(Y <= 1).
    (c) Find P(0.5 < Y < 1.3).
    (d) Find P(Y >= 0.8).
    
    
  • Exercise 4.47 (2 Points):
    Generate two random numbers between 0 and 1 and take Y to be their sum. Then Y is a continuous random variable that can take any value between 0 and 2. The density curve of Y is the triangle shown in Figure 4.13.
    (a) Verify by geometry that the area under this curve is 1.
    (b) What is the probability that Y is less than 1? (Sketch the density curve, shade the area that represents the probability, then find that area. Do this for (c) also.)
    (c) What is the probability that Y is less than 0.5?
    
    
    
    
  • Exercise 4.49 (2 Points):
    An opinion poll asks an SRS of 1500 adults, ``Do you happen to jog?'' Suppose that the population proportion who jog (a parameter) is p = 0.15. To estimate p, we use the proportion phat in the sample who answer ``Yes.'' The statistic phat is a random variable that is approximately normally distributed with mean mu = 0.15 and standard deviation sigma = 0.0092. Find the following probabilities:
    (a) P(phat >= 0.16)
    (b) P(0.14 <= phat <= 0.16)
    
    
  • Exercise 4.52 (3 Points):
    A life insurance company sells a term insurance policy to a 21-year-old male that pays $100,000 if the insured dies within the next 5 years. The probability that a randomly chosen male will die each year can be found in mortality tables. The company collects a premium of $250 each year as payment for the insurance. The amount X that the company earns on this policy is $250 per year, less the $100,000 that it must pay if the insured dies. Here is the distribution of X. Fill in the missing probability in the table and calculate the mean earnings mu_X.

    Age at death	21	22	23	24	25	>= 26
    Payout	-$99,750	-$99,500	-$99,250	-$99,000	-$98,750	$1250
    Probability	0.00183	0.00186	0.00189	0.00191	0.00193	?

    
    
    
  • Exercise 4.53 (3 Points):
    It would be quite risky for you to insure the life of a 21-year-old friend under the terms of the previous exercise. There is a high probability that your friend would live and you would gain $1250 in premiums. But if he were to die, you would lose almost $100,000. Explain carefully why selling insurance is not risky for an insurance company that insures many thousands of 21-year-old men.
    
    
  • Exercise 4.62 (3 Points):
    In an experiment on the behavior of young children, each subject is placed in an area with five toys. The response of interest is the number of toys that the child plays with. Past experiments with many subjects have shown that the probability distribution of the number X of toys played with is as follows:

    Number of toys xi	0	1	2	3	4	5
    Probability pi	0.03	0.16	0.30	0.23	0.17	0.11

    Calculate the mean mu_X and the standard deviation sigma_X.
    
    
• 2) In a previous Stat 2000 class, the following point gains and point losses between Quiz 1 and Quiz 2 have been reported:
  
  

  # Students	Point Gain/Loss
  2	-5
  3	-1
  3	0
  4	+1
  2	+2
  4	+3
  2	+5

  
  Let the random variable Y represent the point gain/loss.
  
  Based on this data, draw a probability histogram, draw a spike graph, determine the cumulative probability function F(y), and draw a graph of F(y). Mark clearly which points are included / not included at particular positions in the graph of F(y). (8 Points)