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Unformatted text preview: AMA527: Decision Analysis Assignment 1 Due 13 October 2010 during lecture 1.(a)[5 pts] A continuous random variable is said to have a gamma distribution with parameters ( t,λ ), λ > 0, and t > 0 if its density function is given by f ( x ) = λe- λx ( λx ) t- 1 Γ( t ) x ≥ x < , where Γ( t ) = Z ∞ e- y y t- 1 dy. Suppose that W , the amount of moisture in the air on a given day, is a gamma random variable with parameters ( t,β ). Suppose also that given that W = w , the number of accidents during that day - call it N- has a Poisson distribution with mean w . Show that the conditional distribution of W given that N = n is the gamma distribution with parameters ( t + n,β + 1). (b)[5 pts] The joint density function of X and Y is given by f ( x,y ) = 1 y e- ( y + x y ) , x > ,y > . Find E ( X ), E ( Y ), and show that V ( X,Y ) = 1. (c)[5 pts] The joint density of X and Y is given by f ( x,y ) = e- x/y e- y y , < x < ∞ ; 0 < y < ∞ ....
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This note was uploaded on 10/12/2010 for the course AMA AMA527 taught by Professor Dr.sim during the Spring '10 term at New School.
- Spring '10