HW1 - Mathematics Department, UCLA T. Richthammer spring...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
Mathematics Department, UCLA T. Richthammer spring 09, sheet 1 Mar 30, 2009 Homework assignments: Math 170B Probability, Sec. 1 01. Let X be a normal RV with parameters μ, σ 2 , and a, b R . (a) Show that Y = aX + b again is a normal RV (with which parameters?) if a ± = 0. (Hint: consider the cases a > 0 and a < 0 separately.) (b) What happens if a = 0? Answer: (a) For a > 0 we have F Y ( c ) = P ( Y c ) = P ( aX + b c ) = P ( X c - b a ) = F X ( c - b a ), so f Y ( c ) = d dc F X ( c - b a ) = f X ( c - b a ) 1 a = 1 2 πσ e - ( c - b a - μ ) 2 / 2 σ 2 1 a = 1 2 πaσ e - ( c - b - ) 2 / 2 a 2 σ 2 , so Y is a normal RV with parameters μ ± = + b and σ ± 2 = a 2 σ 2 . Similarly for a < 0 we have F Y ( c ) = P ( aX + b c ) = P ( X c - b a ) = 1 - F X ( c - b a ), so f Y ( c ) = - d dc F X ( c - b a ) = - f X ( c - b a ) 1 a = 1 2 π | a | σ e - ( c - b - ) 2 / 2 a 2 σ 2 , so again Y is a normal RV with parameters μ ± = + b and σ ± 2 = a 2 σ 2 . (b) For a = 0 we have Y = b , so Y is a discrete RV with all the mass on the value b . 02. Let X 1 , X 2 be independent exponential RVs with parameter 1. Determine the PDF of (a) Y = max { X 1 , X 2 } (b) Z = min { X 1 , X 2 } . Answer: (a) R ( Y ) = (0 , ) For c > 0 we have F Y ( c ) = P ( Y c ) = P (max { X 1 , X 2 } ≤ c ) = P ( X 1 c, X 2 c ) = P ( X 1 c ) P ( X 2 c ) = (1 - e - c ) 2 . So f Y ( c ) = d dc F Y ( c ) = 2(1 - e - c ) e - c , so f Y ( y ) = 2( e - y - e - 2 y )1 { y> 0 } . (b) R ( Z ) = (0 , ) For c > 0 we have 1 - F Z ( c ) = P ( Z > c ) = P (min { X 1 , X 2 } > c ) = P ( X 1 > c, X 2 > c ) = P ( X 1 > c ) P ( X 2 > c ) = e - 2 c . So f Z ( c ) = d dc F Z ( c ) = 2 e - 2 c , so f Z ( z ) = 2 e - 2 z 1 { z> 0 } . 03. The joint PDF of X and Y is given by f ( x, y ) = ( y - x ) e - y 1 { 0 <x<y } . (a) Check that f ( x, y ) is a density function. (b) Find the PDFs of X and Y . Are X and Y independent? (c) Calculate E ( Y ), P ( 1 3 Y X 2 3 Y ) and P ( X + Y 1). Answer: (a) We have f ( x, y ) 0 and f is normalized: R dy R dxf ( x, y ) = R dy R dx ( y - x ) e - y 1 { 0 <x<y } = R 0 dye - y R y 0 dx ( y - x ) = R 0 dye - y y 2 2 = 1. (b) f Y ( y ) = R dxf ( x, y ) = e - y y 2 2 1 { y> 0 } by the same calculation as in (a). f X ( x ) = R dyf ( x, y ) = R x dy ( y - x ) e - y 1 { x> 0 } = R 0 dy ± y ± e - y ± - x 1 { x> 0 } = e - x 1 { x> 0 } , so X is an exponential RV with parameter 1. f ( x, y ) ± = f X ( x ) f Y ( y ), so X and Y are not independent. (c) E ( Y ) = R dyyf Y ( y ) = R 0 dye - y y 3 2 = 3. P ( 1 3 Y X 2 3 Y ) = R 0 dye - y R 2 y/ 3 y/ 3 dx ( y - x ) = R 0 dye - y 1 6 y 2 = 1 3 .
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
P ( X + Y 1) = R dy R dx ( y - x ) e - y 1 { 0 <x<y,x + y 1 } = R 1 0 dye - y R min { y, 1 - y } 0 dx ( y - x ) = R 1 / 2 0 dye - y R y 0 dx ( y - x ) + R 1 1 / 2 dye - y R 1 - y 0 dx ( y - x ) = R 1 / 2 0 dye - y y 2 2 + R 1 1 / 2 dye - y
Background image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 4

HW1 - Mathematics Department, UCLA T. Richthammer spring...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online