Download Randomly Selected - Probablity - Exam and more Exams Probability and Statistics in PDF only on Docsity!
Name:
Math 29 – Probability
Practice Final Exam
Instructions:
- Show all work. You may receive partial credit for partially completed problems.
- You may use calculators and a two-sided sheet of reference notes. You may not use any other references or any texts, except the provided z-table.
- You may not discuss the exam with anyone but me.
- Suggestion: Read all questions before beginning and complete the ones you know best first. Point values per problem are displayed below if that helps you allocate your time among problems.
- You need to demonstrate that you can solve all integrals in problems that do not have a (DO NOT SOLVE) statement. I.E. write out some work showing how you solved the integration, including if necessary integration by parts.
- Probabilities should be given as NUMERICAL values, unless I say an expression is warranted.
- Good luck!
Note: The points total on your final will be 100 points. This was my fast attempt to assign points here, so you can see what I thought the problems were worth before any adjustments.
Problem 1 2 3 4 5 6 7 8 9 Total
Points Earned
Possible Points 10 11 10 10 18 10 10 4 6 90
- Three bowls are labeled 1, 2, and 3, respectively. Bowl i contains i white and 5- i red balls. In an experiment, a bowl is randomly selected from the set of three bowls. Then, 3 balls are randomly selected without replacement from the contents of the selected bowl.
a. Given that bowl 1 was NOT selected, what is the probability of drawing exactly 2 red balls?
b. What is the probability that exactly 2 red balls are drawn?
c. Given that exactly 2 red balls were drawn, what is the probability that bowl 3 was selected?
3. Consider a random variable X with pdf given by f ( x ) kxe x /^4 , x>0, and 0, otherwise.
a. Find the value of k that makes this a valid pdf.
b. Find the mgf for X. You may either identify the distribution and provide its associated mgf or derive the mgf directly.
c. Find the mean and variance of X using the moment generating function you found in b.
4. A company needs a vast amount of iron ore for a project. Suppose X 1 , X 2 ,..., X 40 are a random
sample of measurements on the proportion of impurities in iron ore samples from “Ores R Us” (a supplier company). The proportion of impurities in the population of all similar iron ore samples, X, has
pdf f ( x ) 3 x^2 , 0 x 1 , and 0, otherwise.
a. The company will refuse to buy the ore if X exceeds .8. Find the approximate probability that
X exceeds .8 for a sample of size 40.
b. What numerical value does X 40 converge in probability to? Justify your answer.
b. Show that X and Y are independent and have identical distributions (provide the marginal pdf they share).
c. The two counties want to hire a single company for the repairs. One particular company will only handle combined jobs of at most 6 miles at a time for a given week before charging huge additional fees. Using a probabilistic argument (i.e. compute a meaningful probability), would you recommend the counties use this company for their repairs?
- Suppose X and Y are random variables where Var(X)=8 and Var(Y)=6.
a. If X and Y are independent, what is the variance of 6 X 3 Y 2?
b. If X and Y have correlation .4, what is the variance of X 2 Y?
c. (A Little Theory) If X and Y are independent random variables, show that E ( Y 3 | X ) E ( Y^3 ). You
may treat X and Y as continuous random variables, and use regular notation for their joint pdf and marginal pdfs.
- The class is throwing a celebratory party for the end of the semester. A large number of pizzas are ordered – 40% from Antonio’s and the rest from Domino’s. Of the Domino’s pizzas, 30% are cheese only while the rest have some toppings. From Antonio’s, only 15% are cheese only. What is the probability that a pizza came from Antonio’s if it is known to have toppings besides cheese?
- Matching. (Not all choices may be used.)
_____ A stochastic process where the random variables A. Normal are related by conditional probabilities B. Combination C. Permutation _____ A distribution that may be used to approximate the Poisson D. Markov Chain E. Poisson _____ Result related to convergence in probability F. Cauchy G. Weak LLN _____ Combinatorial method that is employed when H. Gamma order of objects in a subset does matter I. Sample Mean
_____ Example distribution where the mean doesn’t exist
_____ Example distribution where the mean and variance are equal
