Math 135A Final Exam
数学期末考试代写 2.(24 points.) Cards are dealt without replacement from a standard(a)Find the conditional probability that the second card is a club given
NAME(print in CAPITAL letters, first name first):
There are six problems. Make sure that you have all 6 problems. You must show all your work to receive full credit. Do not simplify complicated expressions unless instructed to do so. 数学期末考试代写
1.(24 points.) A pair (X, Y ) of random variables has the jointdensity 数学期末考试代写
(a)Find the value ofc.
(b)Find the marginal densities of X and Y . Are X and Yindependent?
(c)Find P(2Y < X | Y <X). 数学期末考试代写
(d)FindE. X Σ.
2.(24 points.) Cards are dealt without replacement from a standard
(a)Find the conditional probability that the second card is a club given that the first card is club. 数学期末考试代写
(b)Find the conditional probability that the second card is a club given that the second-to- last card is a club.
(d)Find the conditional probability that the first card is a club given that none of the next five cards are clubs.
(d)Find the expected number of clubs that appear before the king of diamonds.
3.(24 points.) A fair die is rolled twice. Let X and Y be the smallest and largest, respectively, number that appears in the two 数学期末考试代写
(a)Determinethe probability mass function of (X, Y ). (Write a formula for P(X = i, Y = j) or give a table of values.)
(b)Are X and Yindependent? 数学期末考试代写
(c)Find E(X + Y ). (Give your answer as a decimal)
- (24 points.) A four-sided die has a 0, two 1’s, and a 2 on its faces. Let X1, X2, . . . be the numbers that appear in successive rolls of this die, and let Sn= X1 + · · Xn.
(a)Find the mean and variance ofX1.
(b)Using a relevant approximation, find the probability that S200> 210.
(c)Find an exact expression for the probability that S200= 200. (Hint: simulate a roll of this die using two fair coins.)
5.(24) In Sweden, 10 people are struck by lightning every year, on average. 数学期末考试代写
Answer the following questions about a given year in Sweden using a relevant approximation. (Assume person-to-person and day-to-day independence, and assume all months have the same length and the same chance of lightning.)
(a)Find the probability that exactly five people are struck by lightning during the year.
(b)Find the probability that a total of two people are struck by lightning during first fourmonths of the year. 数学期末考试代写
(c)Say that a month is good is no one is struck by lightning, and bad Find the probability that the year has 5 good and 7 badmonths.
6.(24) 5 Swedes, 5 Norwegians and 5 Finns are standing in a row in random order.
(a)Find the probability that the leftmost five people are all Swedes.
(b)Saythat a country is happy if all of its citizens are standing Find the probability that Sweden is happy.
(c)Find the expected number of happy countries.
(d)Find the probability that there are no happy countries.