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math330ps4f09

# math330ps4f09 - Assume that the appropriate probability...

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Math 330 Statistics I Problem Set 4 Fall 2009 This assignment will be graded, so you must do your own work. You may use your class notes and text, but no outside references. You may not receive assistance from anyone else. If you have questions about the problems, see me. In order to receive full credit you must show clearly all steps necessary to obtain your answers. Be complete and precise. On the other hand, including irrelevant work may also result in loss of credit. Incoherent and illegible work will not be considered. 1. In the production of a certain type of copper, two types of copper powder (types A and B) are mixed together and sintered (heated) for a certain length of time. For a fixed volume of sintered copper, the producer measures the proportion Y 1 of the volume due to solid copper (some pores will have to be filled with air) and the proportion Y 2 of the solid mass due to type A crystals.
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Unformatted text preview: Assume that the appropriate probability densities for Y 1 and Y 2 are f 1 ( y 1 ) = 6 y 1 (1 − y 1 ), ≤ y 1 ≤ 1, 0, elsewhere f 2 ( y 2 ) = 3 y 2 2 , ≤ y 2 ≤ 1, 0, elsewhere The proportion of the sample volume due to type A crystals is then Y 1 Y 2 . Assume that Y 1 and Y 2 are independent. Find the probability density of U = Y 1 Y 2 . 2. Suppose Y 1 is normally distributed with mean 5 and variance 1 and Y 2 is normally distributed with mean 4 and variance 3. If Y 1 and Y 2 are independent, find P(Y 1 > Y 2 ). 3. If Y is a continuous random variable and m is the median of the distribution, then m is such that P(Y ≤ m ) = P(Y ≥ m ) = 1/2. Let Y 1 , Y 2 , . .., Y n be independent, exponentially distributed random variables with mean β and median m . Find P(Y ( n ) > m ). 1...
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