SuggestedSolPS1 - -λx ´³ 1 -x exp-λx-exp-λx ´ 2...

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Economics/Mathematics C103: Introduction to Mathematical Economics SUGGESTED SOLUTIONS: Problem Set 1 1. Consider the exponential distribution F ( x ) = 1 - exp( - λx ). Assume ω = . (a) Compute the probability density function. The pdf is given by f ( x ) = F 0 ( x ). Therefore, f ( x ) = λ exp( - λx ) (b) Compute its expectation. (The answer is in the book, but of course you are expected to prove why.) Use integration by parts: E [ X ] = lim ω →∞ Z ω 0 xf ( x ) dx = lim ω →∞ Z ω 0 λx exp( - λx ) dx by parts z}|{ = lim ω →∞ - x exp( - λx ) | ω x =0 + Z ω 0 exp( - λx ) dx = lim ω →∞ - ω exp( - λx ) - exp( - λx ) λ ω x =0 = lim ω →∞ ω exp( - λω ) - exp( - λω ) λ + 1 λ = 1 λ (c) Compute the expectation of γ ( x ) = x 2 . E [ γ ( X )] = lim ω →∞ Z ω 0 x 2 f ( x ) dx = lim ω →∞ Z ω 0 λx 2 exp( - λx ) dx = lim ω →∞ - x 2 exp( - λx ) + Z ω 0 2 x exp( - λx ) dx | {z } 2 λ E [ X ] = lim ω →∞ - x 2 exp( - λx ) ω x =0 + 2 λ 2 = 2 λ 2 (d) Compute its hazard ratio. 1
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λ ( x ) f ( x ) 1 - F ( x ) = λ exp( - λx ) exp( λx ) = λ (e) Compute the conditional expectation E [ X | X < x ]. E [ X | X < x ] = 1 F ( x ) Z x 0 tf ( t ) dt = 1 1 - exp( - λx ) - ω exp(
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Unformatted text preview: -λx ) ´³ 1 λ-x exp(-λx )-exp(-λx ) λ ´ 2. Consider the uniform distribution F ( x ) = x/ω on [0 ,ω ]. for some arbitrary ω . (a) Compute the probability density function. f ( x ) = F ( x ) = 1 ω (b) Compute its expectation. E [ X ] = Z ω xf ( x ) dx = Z ω x ω dx = ± x 2 2 ω ² ω x =0 = ω 2 (c) Compute the expectation of γ ( x ) = x 2 . E [ γ ( X )] = Z ω x 2 f ( x ) dx = Z ω x 2 ω dx = ± x 3 3 ω ² ω x =0 = ω 2 3 (d) Compute its hazard ratio. λ ( x ) ≡ f ( x ) 1-F ( x ) = 1 ω 1-x ω = 1 ω-x (e) Compute the conditional expectation E [ X | X < x ]. 2 E [ X | X < x ] = 1 F ( x ) Z x tf ( t ) dt = ω x Z x t 1 ω dt = 1 x ± t 2 2 ω ² x t =0 = ³ 1 x ´³ x 2 2 ω ´ = x 2 3...
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