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Name: Statistics Davidson College Economics 204, Fall 2002 Mark C. Foley
Review # 1
Due by 1:30 p.m. Wednesday, September 25, 2002 in Chambers 202 Directions : This review is closed-book, closed-notes (except for your formula sheet) to be taken in one sitting not to exceed 3 hours. You may use a calculator. Perform your calculations to 3 decimal places. There are 100 points on the exam. Each problem is worth 20 points. You must show all your work to receive full credit. Any assumptions you make and intermediate steps should be clearly indicated. Do not simply write down a final answer to the problems without an explanation. Please turn in your formula sheet with your exam. Carpe diem. Honor Pledge Start time End time
(a) Consider two random variables, X with mean ^ X and variance
2 (^) X , and Y with mean ^ Y and
variance Y^2. Prove Var [c X dY ] c^2 2 X d^2 Y^2 2 cd Cov( X , Y )where c and d are positive
constants. (b) Given
1 2 2
n
x x
s
n i i X , prove
2 1 2 2
n
x n x
s
n i i X
(c) Prove that the random variable X
Z X^ X
has mean 0 and variance 1.
Bert and Ernie have three activities they can choose to do each day. They can either (a) go see Mr. Rogers down the street, (b) play backgammon, or (c) read the New York Times newspaper. The probability they will go see Mr. Rogers is .4, as is the probability they will read the newspaper. The probability they will go see Mr. Rogers and not play backgammon is .3, while the probability that they will play backgammon and not go see Mr. Rogers is .2. The probability they will go see Mr. Rogers and read the newspaper is .1, as is the probability they will play backgammon and read the newspaper. If they read the newspaper, the probability they will go see Mr. Rogers and/or play backgammon is .5. (a) What is the probability they will go see Mr. Rogers and play backgammon? (b) Given that they play backgammon, what is the probability that they will go see Mr. Rogers? (c) What is the probability that, in one day, they will go see Mr. Rogers, play backgammon, and read the newspaper? (d) What is the probability that, in one day, they will not go see Mr. Rogers, not play backgammon, and not read the newspaper?
Assume that savings and income (in thousands of dollars) for a household are jointly distributed random variables with the following joint probability function: Income 0 2 4 Savings -1^ .1^ .05^. 0 0 .1. 1 .1 .15. (a) Are savings and income statistically independent? Show your work. (b) If savings are either 0 or $1 thousand, what is the probability that income is $2 thousand? (c) What is the standard deviation of savings? (d) Calculate and interpret the correlation between savings and income. (e) If expenditure is defined as income minus savings, what is the variance of expenditure?
