Ans a x 1 2 for 0 x 1 b 1 1 3 c 16 d 1 4 12 Suppose P X x x 21 for x 1 2 3 4 5

Ans a x 1 2 for 0 x 1 b 1 1 3 c 16 d 1 4 12 suppose p

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Ans. (a) x 1 / 2 for 0 x 1, (b) 1 - 1 / 3, (c) 1/6, (d) 1 / 4. 12. Suppose P ( X = x ) = x/ 21 for x = 1 , 2 , 3 , 4 , 5 , 6. Find all the medians of this distribution. Ans. 5 is the only median 13. Suppose X has a Poisson distribution with λ = ln 2. Find all the medians of X . Ans. any number in [0 , 1]. 14. Suppose X has a geometric distribution with success probability 1/4, i.e., P ( X = k ) = (3 / 4) k - 1 (1 / 4). Find all the medians of X . Ans. P ( X k ) = (3 / 4) k = 1 / 2 when k = log(1 / 2) / log(3 / 4) = 2 . 409. The only median is 2 since P ( X 2) 1 / 2 and P ( X > 2) < 1 / 2 implies P ( X 2) 1 / 2. 15. Suppose X has density function 3 x - 4 for x 1. (a) Find a function g so that g ( X ) is uniform on (0 , 1). (b) Find a function h so that if U is uniform on (0 , 1), h ( U ) has density function 3 x - 4 for x 1. Ans. (a) 1 - x - 3 (b) (1 - u ) - 1 / 3 . 16. Suppose X 1 , . . . , X n are independent and have distribution function F ( x ). Find the distribution functions of (a) Y = max { X 1 , . . . , X n } and (b) Z = min { X 1 , . . . , X n } Ans. (a) F ( x ) n , (b) 1 - (1 - F ( x )) n 17. Suppose X 1 , . . . , X n are independent exponential( λ ). Show that min { X 1 , . . . , X n } = exponential( ) Functions of random variables 18. Suppose X has density function f ( x ) for a x b and Y = cX + d where c > 0. Find the density function of Y . Ans. f (( y - d ) /c ) when ca + d y cb + d , 0 otherwise 19. Show that if X = exponential(1) then Y = X/λ is exponential( λ ). 20. Suppose X is uniform on (0 , 1). Find the density function of Y = X n . Ans. y (1 /n ) - 1 /n for 0 < y < 1, 0 otherwise 21. Suppose X has density x - 2 for x 1 and Y = X - 2 . Find the density function of Y . 58
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Ans. y - 1 / 2 / 2 22. Suppose X has an exponential distribution with parameter λ and Y = X 1 . Find the density function of Y . This is the Weibull distribution . Ans. αλy α - 1 exp( - λy α ). 23. Suppose X has an exponential distribution with parameter 1 and Y = ln( X ). Find the distribution function of X . This is the double exponential distribution . Ans. 1 - exp( - e - x ) 24. Suppose X is uniform on (0 , π/ 2) and Y = sin X . Find the density function of Y . The answer is called the arcsine law because the distribution function contains the arcsine function. Ans. π - 1 (1 - x 2 ) - 1 / 2 25. Suppose X has density function f ( x ) for - 1 x 1, 0 otherwise. Find the density function of (a) Y = | X | , (b) Z = X 2 . Ans. (a) f ( y ) + f ( - y ) for 0 y 1, (b) { f ( z ) + f ( - z ) } / 2 z for 0 z 1 26. Suppose X has density function x/ 2 for 0 < x < 2, 0 otherwise. Find the density function of Y = X (2 - X ) by computing P ( Y y ) and then differentiating. Ans. f ( y ) = (1 - y ) - 1 / 2 / 2 Joint distributions
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