NotesSep13 - 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

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Unformatted text preview: 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 > ,x 2 > ,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. 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...
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This note was uploaded on 12/03/2010 for the course OR&IE 3500 taught by Professor Samorodnitsky during the Fall '10 term at Cornell University (Engineering School).

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NotesSep13 - 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

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