Probablity 1 Solution, Exercises - Mathematics, Exercises of Probability and Statistics

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2011 Fall 10 ‐ 705 Homework 1 Solutions

2.7 a.

2

1

y

y X

y X

P y X y y f^ x dx^ y P Y y P X y P X y y (^) f x dx y





 ^ ^ ^   

Differentiation gives

Y

y y f y

y y

 ^  

 ^ 

   

     

        

        

 ^   ^   ^ 

2 2

• • 1 2 2

2 2

• • 1 * 1

2

2

2

2 2

2

2

2 *

log log

log (3.31 (b))

exp

j j

j j

i i j i

j

j

j

j j j

c c

c c c

c h x t t

t

t t V

x dx x

t x h x c dx x

x x ar t x



 

 ^  

 ^  

 



3.32 b.

 

 

   

 

1 exp^ /^1 ; , exp 1 ln

x x x x x

  

 ^ ^ 

The natural parameters and sufficient statistics are  [ 1 /  ,  1], t (^)  x (^)  [ , log x x ]. Further,

       

 log c    log  log    log 1 /  1  log

Therefore

     

1 1 1 1

x log c log 1/