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# Quiz2-Answer - Probability Quiz II 1 A lling station is...

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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 0 < x < 1 0 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 0 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 - 0 . 01 1 / 5 0 . 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 ) , 0 x < , 0 y < . (a) Find P ( X Y ). (5%) Sol) P ( X Y ) = Z 0 Z x e - ( x + y ) dydx = Z 0 e - x Z x e - y dydx = Z 0 e - 2 x dx = 1 2 (b) Find P ( X a ). (5%) Sol) f X ( x ) = Z 0 e - ( x + y ) dy = e - x , x > 0 P ( X a ) = Z a 0 e - x dx = 1 - e - a , a > 0 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 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 . 1

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(a) Find the value of c such that f ( x, y ) is a legitimate joint density function. (5%) Sol) 1 = Z 0 Z 0 f ( x, y ) dydx = c Z 1 0 x Z x 0 ydydx = c 2 Z 1 0 x 3 dx = c 8 = c = 8 (b) Find P ( Y > 1 - X ). (5%) Sol) In the following figure, the darker area satisfies condition Y > 1 - X .
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