FINAL EXAM
STAT 115 A SPRING 24
Total Points 100
1. 统计期末代考
Prove the following properties of covariance: (15 pts)
- cov(X, Y) = cov(Y, X)
- cov(X,X) = Var(X)
- cov(aX, Y) = a cov(X, Y) for any constant a
- cov(X+c, Y+d) = cov(X,Y) where cand d are constants
2.
Suppose you are modeling the number of accidents occurring at a particular intersection each month. Which distribution would you use for this scenario? Justify your choice. (5 pts)
3. 统计期末代考
Let Xbe a random variable with the following probability mass function: (15 pts)
P(X = k) = 1/2k for k = 1, 2, 3,…
Compute the moment generating function MX(t) for X, and hence find the mean and variance of X.
4.
Suppose you are waiting for a bus at a bus stop, and the waiting time until the arrival of the next bus follows an exponential distribution with a mean waiting time of 10 minutes. You have already waited for 5 minutes.
What is the probability that you will have to wait for more than 10 minutes in total before the next bus arrives? (5 pts)
5. 统计期末代考
You are part of a research team investigating a rare genetic disorder affecting 1 in 10,000 individuals in a population. A new diagnostic test has been developed to detect the disorder. The test has a sensitivity of 98% (the probability of a positive test result given that the individual has the disorder) and a specificity of 99% (the probability of a negative test result given that the individual does not have the disorder).
a. Given that an individual tests positive for the disorder, what is the probability that they actually have the disorder? (5 pts)
b. Given that an individual tests negative for the disorder, what is the probability that they do not have the disorder? (5 pts)
6. 统计期末代考
You are tasked with explaining the concept of the standard normal distribution to a group of students who are learning about probability distributions for the first time. How would you explain in no less than 100 words what the standard normal distribution is and why it is important in statistics? (10 pts)
9. 统计期末代考
A company offers a basic life insurance policy to its employees, as well as a supplemental life insurance policy. To purchase the supplemental policy, an employee must first purchase the basic policy. Let X denote the proportion of employees who purchase the basic policy, and Y the proportion of employees who purchase the supplemental policy. Let X and Y have the joint density function f(x,y) = 2(x + y) on the region where the density is positive. Given that 10% of the employees buy the basic policy, what is the probability that fewer than 5% buy the supplemental policy? (15 pts)
10.
From 27 pieces of luggage, an airline luggage handler damages a random sample of four. The probability that exactly one of the damaged pieces of luggage is insured is twice the probability that none of the damaged pieces are insured. Calculate the probability that exactly two of the four damaged pieces are insured. (5 pts)
