STAT 400 Lecture AL1 Practice Problems 4 (Part 2)Spring 2015 Dalpiaz 1. Let X and Y have the joint probability density function f X, Y ( x, y ) = 2 163yx, 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 ). c) Find P ( X + Y > K ). d) Are X and Y independent? 2. Let S and T have the joint probability density function f S , T ( s, t ) = t1, 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 ) = yxC , 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 otherwise010 ,10 ,2yxyxkyxf a) Find the value of k that would make yxf, 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 ¼ , 0 Y ¼ ). c) What are the marginal probability density functions of X and Y? Are X and Y independent? d) Find P( ¼ Y ¾ ). 5. Let X and Y be two independent random variables, with probability density functions f X ( x ) and f Y ( y ) , respectively. f X ( x ) = xex , x > 0, f Y ( y ) = ye , 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. x 2 0 1 2 3 x 1 0 0.08 0.07 0.04 0.00 1 0.06 0.15 0.05 0.04 2 0.05 0.04 0.10 0.06 3 0.00 0.03 0.04 0.07 4 0.00 0.01 0.05 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 otherwise002 2X xxfxe and Y has probability density function otherwise003 3Y yyfye a) What is the joint probability density function of ( X , Y ) ? b) Find P( X < Y ). c) Find P( X > ½ ), P( Y > ½ ), P( X > ½ Y > ½ ), P( X > ½ Y > ½ ).1. Let X and Y have the joint probability density function f X, Y ( x, y ) = 2 163yx, 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. 1 = KKdydxyx002 163 = 325 K. K = 2. b) Find P ( Y > 2 X ). P ( Y > 2 X ) = 202 02 163 dydxyxy = 204 1283 dyy = 203 = 0.15. OR P ( Y > 2 X ) = 10222 163 dxdyyxx = … = 203 = 0.15. c) Find P ( X + Y > K ). P ( X + Y > 2 ) = 20222 163 dydxyxy = 20 2 2 24 323 dyyy= 2043 4 323 dyyy = 02 54 51323 yy = 53216323 = 0.90. OR P ( X + Y > 2 ) = 20202 163 1 dydxyxy = 202 2 2 323 1 dyyy = 20432 44 3231 dyyyy = 02 543 51343231 yyy = 532163323231 = 0.90. d) Are X and Y independent? f X ( x ) = 202 163 dyyx = x 21, 0 < x < 2, f Y ( y ) = 202 163 dxyx = 2 83y , 0 < y < 2. f ( x, y ) = f X ( x ) f Y ( y ). 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. Let S and T have the joint probability density function f S , T ( s, t ) = t1, 0 < s < 1, s 2 < t < s. a) Find f S ( s ) and f T ( t ). f S ( s ) = ssdtt2 1 = 2 lnsst = ln s – ln s 2 = – ln s, 0 < s < 1. f T ( t ) = ttdst 1 = 1 ttt = 11 t, 0 < t < 1. b) Find E ( S ) and E ( T ). E ( S ) = 10 ln dsss = 0122 4 2 lnsss = …
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