This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Î»x ) Â´Â³ 1 Î»x exp(Î»x )exp(Î»x ) Î» Â´ 2. Consider the uniform distribution F ( x ) = x/Ï‰ on [0 ,Ï‰ ]. for some arbitrary Ï‰ . (a) Compute the probability density function. f ( x ) = F ( x ) = 1 Ï‰ (b) Compute its expectation. E [ X ] = Z Ï‰ xf ( x ) dx = Z Ï‰ x Ï‰ dx = Â± x 2 2 Ï‰ Â² Ï‰ x =0 = Ï‰ 2 (c) Compute the expectation of Î³ ( x ) = x 2 . E [ Î³ ( X )] = Z Ï‰ x 2 f ( x ) dx = Z Ï‰ x 2 Ï‰ dx = Â± x 3 3 Ï‰ Â² Ï‰ x =0 = Ï‰ 2 3 (d) Compute its hazard ratio. Î» ( x ) â‰¡ f ( x ) 1F ( x ) = 1 Ï‰ 1x Ï‰ = 1 Ï‰x (e) Compute the conditional expectation E [ X  X < x ]. 2 E [ X  X < x ] = 1 F ( x ) Z x tf ( t ) dt = Ï‰ x Z x t 1 Ï‰ dt = 1 x Â± t 2 2 Ï‰ Â² x t =0 = Â³ 1 x Â´Â³ x 2 2 Ï‰ Â´ = x 2 3...
View
Full Document
 Spring '09
 Anh
 Math, Probability, Probability theory, lim, lim exp, lim x2 exp

Click to edit the document details