NotesSep13

NotesSep13 - Example When a certain method is used to...

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Example When a certain method is used to collect a fixed volume of rock samples in a region, there are 4 rock types. Let X 1 , X 2 and X 3 be the proportions by volume of the first 3 rock types in randomly selected sample. Based on empirical data, X 1 , X 2 , X 3 are mod- eled as a continuous random vector with a joint pdf f X 1 ,X 2 ,X 3 ( x 1 , x 2 , x 3 ) = 144 x 1 x 2 (1 - x 3 ) for x 1 > 0 , x 2 > 0 , x 3 > 0 and 0 < x 1 + x 2 + x 3 < 1. Find the probability that rocks of type 1 and 2 together account for at most 50% of the volume of the sample.

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Independent Random Variables Independent random variables are such that, knowing the value of one of them, does not tell us anything about the values of the rest. Formally, random variables X 1 , . . . , X n are in- dependent if for all sets A 1 , . . . , A n P ( X 1 A 1 , . . . , X n A n ) = P ( X 1 A 1 ) . . . P ( X n A n ) = n Y j =1 P X j A j .
Criteria for independence Random variables X 1 , . . . , X n are indepen- dent if and only if the joint cdf factors out into a product of the marginal cdfs: F X 1 ,...,X n ( x 1 , . . . , x n ) = P ( X 1 x 1 , . . . , X n

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• '10
• SAMORODNITSKY
• Probability theory, Xn

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