STAT 400 Lecture AL1 1 Practice Problems 4 Part 2 Spring 2015 Dalpiaz Let X and Y have the joint probability density function 3 x y2 16 f X Y x y 0 x K 0 y K a Find the value of K so that f X Y x y is a valid joint p d f b Find P Y 2 X d Are X and Y independent 2 Let S and T have the joint probability density function f S T s t Find P X Y K c 1 t 0 s 1 s 2 t s a Find f S s and f T t b Find E S and E T c Find the correlation coefficient S T 3 Let the joint probability density function for X Y be f x y C x y x 0 y 0 x 2 y 3 2 25 zero elsewhere a Find the value of C so that f x y is a valid joint p d f b Find P 2 X Y 2 c Find P X 3 Y 0 4 A bank operates both a drive up facility and a walk up window On a randomly selected day let X the proportion of time that the drive up facility is in use at least one customer is being served or waiting to be served and Y the proportion of time that the walk up window is in use Then the set of possible values for X Y is the rectangle D x y 0 x 1 0 y 1 Suppose the joint probability density function of X Y is given by 2 f x y k x y a 0 0 x 1 0 y 1 otherwise Find the value of k that would make f x y a valid probability density function b Find the probability that neither facility is busy more than one quarter of the time That is find P 0 X 0 Y c What are the marginal probability density functions of X and Y Are X and Y independent d Find P 5 Let X and Y be two independent random variables with probability density Y functions f X x and f Y y respectively fX x x e x x 0 fY y e y y 0 Find the probability P X Y 6 Let X and Y have the joint probability density function f X Y x y x x 0 0 y e x zero elsewhere a Find f X x and f Y y b Find E X and E Y c Are X and Y independent 7 Let X 1 denote the number of customers in line at the express checkout and X 2 denote the number of customers in line at the regular checkout at a local market Suppose the joint probability mass function of X 1 and X 2 is as given in the table below x2 0 1 2 3 4 x1 0 1 2 3 0 08 0 06 0 05 0 00 0 00 0 07 0 15 0 04 0 03 0 01 0 04 0 05 0 10 0 04 0 05 0 00 0 04 0 06 0 07 0 06 a Find P X 1 X 2 that is find the probability that the number of customers in the two lines are identical b Find the probability that there are at least two more customers in one line than in the other line c What is the probability that the total number of customers in the two lines is exactly four At least four d Determine the marginal probability mass functions of X 1 and X 2 e Are X 1 and X 2 independent 8 Let X and Y be two independent random variables Suppose X has probability density function 2 x x 0 f X x 2 e 0 otherwise and Y has probability density function 3 y 0 f Y y 3 e y 0 otherwise a What is the joint probability density function of X Y b Find P X Y c Find P X P Y P X Y P X Y 1 Let X and Y have the joint probability density function f X Y x y a 3 x y2 16 0 x K 0 y K Find the value of K so that f X Y x y is a valid joint p d f KK 1 b 3 K5 2 x y dx dy 16 32 00 K 2 Find P Y 2 X 3 2 x y dx 16 dy 0 0 2 y 2 P Y 2 X 2 3 4 128 y dy 0 3 0 15 20 OR P Y 2 X c 1 2 3 2 dx 3 0 15 x y dy 16 20 0 2 x Find P X Y K 3 2 P X Y 2 x y dx dy 16 0 2 y 2 2 2 0 3 2 y 4 2 y 2 dy 32 2 3 4 y 3 y 4 dy 32 0 3 4 1 5 y y 32 5 2 0 3 32 16 32 5 0 90 OR 2 2 y 2 3 3 2 2 x y dx dy 1 P X Y 2 1 y 2 y 2 dy 16 32 0 0 0 2 1 3 4 1 3 4 y 2 4 y 3 y 4 dy 1 y 3 y 4 y 5 32 3 5 32 0 1 d 3 32 32 16 0 90 32 3 5 Are X and Y independent 2 fX x 3 16 x y 2 dy 1 x 2 dx 3 2 y 8 0 2 fY y 3 16 x y 2 0 f x y f X x f Y y 0 x 2 0 y 2 X and Y are independent OR The support of X Y is a rectangle f X Y x y can be written as a product of two functions one of x only the other of y only X and Y are independent 2 0 2 Let S and T have the joint probability density function f S T s t a 1 t 0 s 1 s 2 t s Find f S s and f T t s s f S s dt ln t 2 ln s ln s 2 ln s s 2 t 1 0 s 1 s t f T t ds t t t t t b 1 1 t 1 0 t 1 Find E S and E T s2 s2 s ln s ds ln s 2 4 0 1 E S 1 E T 1 1 dt t 0 1 t 0 c 1 1 1 0 4 t t dt 2 1 1 3 2 6 Find the correlation coefficient S T 1 s 1 s 1 1 1 1 E S T s t dt ds s dt ds s 2 s 3 ds 3 4 12 t 2 2 0 s 0 s 0 1 Cov S T …
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