Unformatted text preview:

STOR 435 May 6 2009 Name Final Exam Quick Key All problem parts have equal weight In budgeting your time expect that some problems will take longer than others The exam is closed book and closed notes You will only need a calculator a pen or pencil and one sheet of paper with class material that was prepared by you and does not contain any solutions to homework and old exams If you need additional blank paper for solutions or work please ask the instructor To receive any or partial credit for a question you must show your work Please give your answers in the space provided All numerical answers should be expressed as decimal numbers rather than fractions When you have nished please sign the following Honor Code pledge I pledge that I have neither given nor received unauthorized assistance during this examination Signed 1 You are dealt 5 cards from a regular deck of cards a What is the probability that all your cards are of the same color Recall there is 26 red and 26 black cards in the deck b What is the probability that all your cards are of the same color given that you have exactly two aces Solution a b 2 26 5 52 5 2 24 2 2 3 4 2 48 3 0 051 0 039 2 Joint density of X and Y denoted by fXY x y is given below In all three cases nd the constant c and decide whether X and Y are independent Give reasons a fXY x y c x y b fXY x y c x xy c fXY x y cxy 0 x 1 0 y 2 0 x 1 0 y 2 0 x 1 0 y 2x Solution a c 1 3 not independent the joint density is not the product of b c 1 2 independent the joint density can be factored as a product marginal densities of x 1 y c c 2 not independent the joint density is not the product of marginal densities notice that the bound on y depends on x 3 Suppose that the distribution function of a random variable X is given by 0 0 2 0 8 1 FX s s 1 1 s 0 0 s 1 1 s a Find the probability mass function pX s b Find EX and Var X Solution x 1 0 2 0 0 6 1 0 2 pX x a b EX 1 0 2 0 0 6 1 0 2 0 EX 2 1 2 0 2 02 0 6 12 0 2 0 4 Var X 0 4 02 0 4 4 Let the condition distribution of Y X x be N x 1 and X N 0 1 a Find EY Var Y and X Y b What is the conditional density fX Y y x Is this a distribution you recognize c What is the distribution of X 2 d Find the moment generating function of X 2 a EX 0 Var X 1 EX 2 1 1 1 2 E Y X x fX x dx cid 82 E Y 2 X x fX x dx cid 82 cid 82 xyfY X x y fX x dy dx cid 82 fY X x y fX x dx cid 82 1 2 x2fX x dx 1 Solution EY cid 82 EY 2 cid 82 EY 2 cid 82 cid 82 b fY y cid 82 X t cid 82 y2 4 4 e X Y 1 0 1 1 2t 1 2 2 e y x 2 cid 82 N 0 2 c We have shown in class If X N 0 1 then X 2 2 d M 2 2 1 2 t 1 2z 1 2e z 1 2 t 1 1 e z 2 2 1 1 1 2 2z 1 2 dz 1 2t 1 2 0 ezt 1 2 0 dz xfX x dx EX 0 x2 1 fX x dx cid 16 cid 82 cid 17 x yfY X x y dy fX x dx 2 x2 2 dx e y2 4 4 cid 82 e x y 2 2 dx 5 The annual rainfall in cm in a certain area is normally distributed with 40 and 4 Assume independence between years a What is the probability that starting with next year it will take over 10 years before a year occurs having rainfall over 50 cm b What is the probability that the average rainfall in the next 10 years will be between 37 and 43 cm c What is the probability that next year rainfall is more than 90 of the rainfall in the following year Solution a P annual rainfall less than 50cm next ten years P X 50 10 P Z 50 40 4 10 0 993810 0 9397 b P 37 X 43 P 37 40 c Notice X1 0 9X2 N 40 36 16 0 92 16 N 4 28 96 Thus 0 7704 P X1 0 9X2 P X1 0 9X2 0 P Z 0 4 Z 43 40 0 9821 10 10 4 4 28 96 6 The number of years a certain appliance works is a r v with a haz ard rate function given by t 1 t 2 1 time t is measured in Solution P X 6 1 e R 6 years Find the probability that the appliance will fail within 6 years 0 s ds 1 e 48 100 0 3821 100


View Full Document

UNC-Chapel Hill STOR 435 - Final Exam

Download Final Exam
Our administrator received your request to download this document. We will send you the file to your email shortly.
Loading Unlocking...
Login

Join to view Final Exam and access 3M+ class-specific study document.

or
We will never post anything without your permission.
Don't have an account?
Sign Up

Join to view Final Exam and access 3M+ class-specific study document.

or

By creating an account you agree to our Privacy Policy and Terms Of Use

Already a member?