4.68 Find the mean of the random variable. A random variable X has the following distribution.

Find the mean for this random variable. Show your work.

4.69 Explain what happens when the sample size gets large. Consider the following scenarios: (1) You take a sample of two observations on a random variable and compute the sample mean, (2) you take a sample of 100 observations on the same random variable and compute the sample mean, (3) you take a sample of 1000 observations on the same random variable and compute the sample mean. Explain in simple language how close you expect the sample mean to be to the mean of the random variable as you move from Scenario 1 to Scenario 2 to Scenario 3.

4.70 Find some means. Suppose that X is a random variable with mean 30 and standard deviation 4. Also suppose that Y is a random variable with mean 50 and standard deviation 8. Find the mean of the random variable Z for each of the following cases. Be sure to show your work.

4.71 Find the variance and the standard deviation. A random variable X has the following distribution.

Find the variance and the standard deviation for this random variable. Show your work.

4.72 Find some variances and standard deviations. Suppose that X is a random variable with mean 30 and standard deviation 4. Also suppose that Y is a random variable with mean 50 and standard deviation 8. Assume that the correlation between X and Y is zero. Find the variance and the standard deviation of the random variable Z for each of the following cases. Be sure to show your work.

4.73 What happens if the correlation is not zero? Suppose that X is a random variable with mean 30 and standard deviation 4. Also suppose that Y is a random variable with mean 50 and standard deviation 8. Assume that the correlation between X and Y is 0.5. Find the mean of the random variable Z for each of the following cases. Be sure to show your work.

4.74 What’s wrong? In each of the following scenarios, there is something wrong. Describe what is wrong and give a reason for your answer.

4.75 Servings of fruits and vegetables. The following table gives the distribution of the number of servings of fruits and vegetables consumed per day in a population.

Find the mean for this random variable.

4.76 Mean of the distribution for the number of aces. In Exercise 4.54 (page 244) you examined the probability distribution for the number of aces when you are dealt two cards in the game of Texas hold ’em. Let X represent the number of aces in a randomly selected deal of two cards in this game. Here is the probability distribution for the random variable X:

Find μ_X, the mean of the probability distribution of X.

4.77 Standard deviation of the number of aces. Refer to Exercise 4.76. Find the standard deviation of the number of aces.

4.78 Standard deviation for fruits and vegetables. Refer to Exercise 4.75. Find the variance and the standard deviation for the distribution of the number of servings of fruits and vegetables.

4.79 Suppose that the correlation is zero. Refer to Example 4.38 (page 259).

4.80 Find the mean of the sum. Figure 4.12 (page 245) displays the density curve of the sum Y = X₁ + X₂ of two independent random numbers, each uniformly distributed between 0 and 1.

4.81 Calcium supplements and calcium in the diet. Refer to Example 4.39 (page 260). Suppose that people who have high intakes of calcium in their diets are more compliant than those who have low intakes. What effect would this have on the calculation of the standard deviation for the total calcium intake? Explain your answer.

4.82 Toss a four-sided die twice. Role-playing games like Dungeons & Dragons use many different types of dice. Suppose that a four-sided die has faces marked 1, 2, 3, and 4. The intelligence of a character is determined by rolling this die twice and adding 1 to the sum of the spots. The faces are equally likely, and the two rolls are independent. What is the average (mean) intelligence for such characters? How spread out are their intelligences, as measured by the standard deviation of the distribution?

4.83 Means and variances of sums. The rules for means and variances allow you to find the mean and variance of a sum of random variables without first finding the distribution of the sum, which is usually much harder to do.

4.84 What happens when the correlation is 1? We know that variances add if the random variables involved are uncorrelated (ρ = 0), but not otherwise. The opposite extreme is perfect positive correlation (ρ = 1). Show by using the general addition rule for variances that, in this case, the standard deviations add. That is, if ρ_XY = 1.

4.85 Will you assume independence? In which of the following games of chance would you be willing to assume independence of X and Y in making a probability model? Explain your answer in each case.

4.86 Transform the distribution of heights from centimeters to inches. A report of the National Center for Health Statistics says that the heights of 20-year-old men have mean 176.8 centimeters (cm) and standard deviation 7.2 cm. There are 2.54 centimeters in an inch. What are the mean and standard deviation in inches?

4.87 Fire insurance. An insurance company looks at the records for millions of homeowners and sees that the mean loss from fire in a year is μ = $300 per person. (Most of us have no loss, but a few lose their homes. The $300 is the average loss.) The company plans to sell fire insurance for $300 plus enough to cover its costs and profit. Explain clearly why it would be stupid to sell only 10 policies. Then explain why selling thousands of such policies is a safe business.

4.88 Mean and standard deviation for 10 and for 12 policies. In fact, the insurance company sees that in the entire population of homeowners, the mean loss from fire is μ = $300 and the standard deviation of the loss is σ = $400. What are the mean and standard deviation of the average loss for 10 policies? (Losses on separate policies are independent.) What are the mean and standard deviation of the average loss for 12 policies?