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Unformatted text preview: Probability: Quiz II May 25, 2004 1. A filling station is supplied with gasoline once a week. Suppose its weekly volume of sales in thousands of gallons is a random variable with probability density function f ( x ) = ‰ 5(1 x ) 4 < x < 1 otherwise How large must the capacity of the tank be so that the probability of the supply’s being exhausted in a given week is 0.01? (5%) Sol) We want to find c such that P ( X > c ) = 1 F ( c ) = 0 . 01. For 0 < x < 1, we have F ( x ) = Z x 5(1 t ) 4 dt = 1 (1 x ) 5 Solving the following equation: 1 F ( c ) = 1 [1 (1 c ) 5 ] = 0 . 01 , we get c = 1 . 01 1 / 5 ≈ . 601. Hence, the tank capacity is required to be 601 gallons. 2. The joint probability density function of X and Y is given by f ( x,y ) = e ( x + y ) , ≤ x < ∞ , ≤ y < ∞ . (a) Find P ( X ≤ Y ). (5%) Sol) P ( X ≤ Y ) = Z ∞ Z ∞ x e ( x + y ) dydx = Z ∞ e x Z ∞ x e y dydx = Z ∞ e 2 x dx = 1 2 (b) Find P ( X ≤ a ). (5%) Sol) f X ( x ) = Z ∞ e ( x + y ) dy = e x , x > P ( X ≤ a ) = Z a e x dx = 1 e a , a > 3. Let T be the region bounded by the lines y = 0, y = x , and x = 1, as was shown in the following figure. y = 0 x =1 y = x x y y = 0 x =1 y = x x y Suppose that f is a function defined by f ( x,y ) = cxy for ( x,y ) in T , and that f ( x,y ) = 0 when ( x,y ) is not in T ....
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This note was uploaded on 12/22/2011 for the course CS 101 taught by Professor Dat during the Spring '11 term at Bilkent University.
 Spring '11
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