The optimal scheduling of preventative maintenance tests of some (but not all) of n independently operating components was developed. The time (in hours) between failures of a component was approximated by an exponentially distributed random variable with mean 1700 hours. [Round to 4 decimal places where necessary.]

Find the probability that the time between a component failures ranges is at least 1700 hours.

Find the probability that the time between a component failures ranges between 1700 and 2200 hours.

Suppose a component is still working after 1700 hours, find the conditional probability that it will fail before 2200 hours.

Suppose 8 components are tested. What is the probability that 1 of them failed between 1700 and 2200 hours?

If 8 components are tested, find the probability that at least 1 of them failed between 1700 and 2200 hours

The joint probability distribution of variables X and Y is shown in the table below.

...............................................................................X.......................................................................

1. Calculate E(XY)

2. Determine the marginal probability distributions of X and Y.

3. Calculate E(X) and E(Y)

4. Calculate V(X) and V(Y)

5. Are X and Y independent? Explain.

6. Find P(Y = 2| X = 1)

7. Calculate COV(X,Y). Did you expect this answer? Why?

8. Find the probability distribution of the random variable X + Y.     

i.   Calculate E(X + Y) directly by using the probability distribution of X + Y.

1. Calculate V(X + Y) directly by using the probability distribution of X + Y, and verify that V(X + Y) = V(X) + V(Y). Did you expect this result? Why?

For exercises 21-23, construct a probability distribution and compute the mean and standard deviation for only 21 and 23.

21 Kathryn and John would like to rent a car for a day at the airport. There is a 0.30 probability that they will rent a truck at $20per day, a 0.27 probability that they will rent a SUV at $18 per day, and a 0.28 probability that they will rent a sport car at $35 per day, and a 0.15 probability that they will rent minivan at $24 per day. Let X be random variable be defined as renting cost.

22 A car dealer bought a used car at $2,800. He estimates that he can sell the car for $3,500, $3,700, or $3,900, with probabilities .24, .40, and .36, respectively. Let X be   a random variable defined as profit obtained by selling a car.

23 The probabilities that a customer selects 1, 2, 3, 4, or 5 items at a convenience store are 0.32, 0.12, 0.23, 0.18, and 0.15, respectively

A population has a mean of 200 and a standard deviation of 80. Suppose a sample of size 125 is selected and is used to estimate . Use z-table.

1. What is the probability that the sample mean will be within +/- 5 of the population mean (to 4 decimals)? (Round z value in intermediate calculations to 2 decimal places.)
   
   
2. What is the probability that the sample mean will be within +/- 14 of the population mean (to 4 decimals)? (Round z value in intermediate calculations to 2 decimal places.)

In the EAI sampling problem, the population mean is $51,500 and the population standard deviation is $5,000. When the sample size is n = 30, there is a 0.4161 probability of obtaining a sample mean within +/- $500 of the population mean. Use z-table.

1. What is the probability that the sample mean is within $500 of the population mean if a sample of size 60 is used (to 4 decimals)?
   
   
2. What is the probability that the sample mean is within $500 of the population mean if a sample of size 120 is used (to 4 decimals)?

According to the article "A Critical Appraisal of 98.6 Degrees F, the Upper Limit of the Normal Body Temperature, and Other Legacies of Carl Reinhold August Wunderlich" published in the Journal of the American Medical Association, the body temperatures of adults are normally distributed with a mean of 98.242 and a standard deviation of 0.713.

1) What is the probability that a randomly selected person's body temperature is between 97.9 and 98.51? Round your answer to 4 decimal places.
2) What is the probability that the average body temperature of 4 randomly selected people is between 97.9 and 98.51? Round your answer to 4 decimal places.

3) Why did the probability increase?

• The probability increased since the sample size decreased resulting in a wider distribution which increased the chances of being in an interval containing the mean.
• The probability increased since the sample size increased resulting in a narrower distribution which increased the chances of being in an interval containing the mean.

1. The SAT test scores have an average value of 1200 with a standard deviation of 100. A random sample of 35 scores is selected for study.

A) What is the shape, mean(expected value) and standard deviation of the sampling distribution of the sample mean for samples of size 35?

B) What is the probability that the sample mean will be larger than 1235?

C) What is the probability that the sample mean will fall within 25 points of the population mean?

D) What is the probability that the sample mean will be less than 1180?

2. Assume that the population proportion of adults having a college degree is 0.44. A random sample of 375 adults is to be selected to test this claim.

A) What are the shape, mean, and standard deviation of the sampling distribution of the sample proportion for samples of 375?

B) What is the probability that the sample proportion will be less than 0.50?

C) What is the probability that the sample proportion will fall within 4 percentage points(+/- 0.04) of the population proportion?

1. Length (in days) of human pregnancies is a normal random variable (X) with mean 266, standard deviation 16.

a. The probability is 95% that a pregnancy will last between what 2 days? (Remember your empirical rule here)

b. What is the probability of a pregnancy lasting longer than 315 days?

2. What is the probability that a normal random variable will take a value that is less than 1.05 standard deviations above its mean? In other words, what is P(Z < 1.05)?

3. What is the probability that a normal random variable will take a value that is between 1.5 standard deviations below the mean and 2.5 standard deviations above the mean? In other words, what is P(−1.5 < Z < 2.5)?

4. What is the probability that a normal random variable will take a value that is more than 2.55 standard deviations above its mean? In other words, what is P(Z > 2.55)?

In TaxLand, income tax rate is 30%. Cheolsoo, who is a resident of TaxLand and earns 10 a
year, is going to report his income.
(a) If he underreports his income, it is detected with 10% probability and the penalty is the square of the underreported income. (If he reports 8 and is detected, he should pay
2*2 = 4 as penalty.) If Cheolsoo is risk neutral, how much income is he going to report?
(b) Tax office can increase the detection probability to 20% with the cost of x. Compare the expected tax revenue and penalty income of the tax office. For which range of x, is
it beneficial for the tax office to increase the detection probability?
(c) Suppose the detection probability of underreport is increasing with the amount of un-derreporting. The detection probability is given as 10% + 10% - M, where M is the amount of underreporting. The penalty, once detected, is the same as the amount of underreporting. If Cheolsoo is risk neutral, how much income is he going to report?

The risk of females experiencing an anxiety disorder during a given 12-month period is approximately 1 in 5. Suppose a researcher plans to take a random sample of females and monitor their anxiety over 12 months.

If 20 females are randomly sampled, what is the probability that exactly 10 will experience an anxiety disorder during this 12-month period? (Round answer to 3 decimal places)

If 20 females are randomly sampled, what is the probability that exactly 5 will experience an anxiety disorder? (Round answer to 3 decimal places)  

If 30 females are randomly sampled, what is the probability that exactly 5 will experience an anxiety disorder? (Round answer to 3 decimal places)  

If 20 females are randomly sampled, what is the probability that 5 or 6 will experience an anxiety disorder? (Round answer to 3 decimal places)

If 10 females are randomly sampled, what is the probability that 5 or more will experience an anxiety disorder? (Round answer to 3 decimal places)

Our most accurate information about the current pandemic comes from Italy, where there are 101,739 confirmed cases out of a population of 60 million. This means about 0.17% of the entire population are confirmed cases. Suppose you have 30 extended family members in Italy and each one independently has a 0.17% chance of being a confirmed case. Use binompdf and binomcdf to find the following probabilities:

4a. What is the probability that exactly 0 of your extended family members is a confirmed case?

4b. What is the probability that exactly 10 of your extended family members are confirmed cases?

4c. What is the probability that less than 5 of your extended family members are confirmed cases?

4d. What is the probability that more than 1 of your extended family members are confirmed cases?

4e. What is the probability that between 2 and 4 of your extended family members are confirmed cases?