Probability and Statistics for Engineering and the Sciences (with CD-ROM and InfoTrac )

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Midterm I Solutions Problem 1 Since f is a density function dx x f ) ( = 1. That is dx x f ) ( = 0 ) ( dx x f + 1 0 ) ( dx x f + 1 ) ( dx x f = + 1 0 2 ) ( dx bx a = = + 1 0 3 1 0 3 | x b ax 1 3 = + b a 3 1 b a = On the other hand, EX= 5 3 4 2 4 2 ) ( ) ( 1 0 4 1 0 2 1 0 3 = + = + = + = b a x b x a dx bx ax dx x xf Replacing a by 3 1 b , we get 5 3 4 2 3 1 = + b b 5 6 = b and 5 3 = a Problem 2 Let X= the number of functioning components in a 2-out-of-3 circuit 1. n= 3 components 2. two possible outcomes: the component function or not 3. the trials are independent since each component functions independently 4. the probability of success p=0.9 is constant from trial to trial Therefore X~ Bin(n=3, p=0.9) P(circuit functions)= P(X 2)= P(X=2)+P(X=3)= ) 9 . 0 1 ( ) 9 . 0 ( 2 3 2 + 0 3 ) 9 . 0 1 ( ) 9 . 0 ( 3 3 = 0.243+ 0.729= 0.972 Problem 3 a. Let X= the number of defective screws in a package 1. n= 10 screws 2. two possible outcomes: the screw is defective or not 3. the trials are independent since it is know that the screws are defective independently
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